Doctor of Philosophy Thesis - QUTeprints.qut.edu.au/16497/1/Mark_Glickman_Thesis.pdf · Doctor of Philosophy Thesis Disturbance Monitoring in Distributed Power Systems Mark Glickman,

Doctor of Philosophy Thesis

Disturbance Monitoring in Distributed Power Systems

Mark Glickman, [email protected]

Principal Supervisor: Professor Peter J. O’Shea Associate Supervisor: Professor Gerald F. Ledwich

Second Associate Supervisor: Dr. Edward W. Palmer

School of Engineering Systems (ES) Faculty of Built Environment and Engineering (BEE)

Queensland University of Technology (QUT)

mailto:[email protected]

Statement of Sources

This thesis makes an original contribution to knowledge. The material presented

apart from published papers was not submitted in any form before. Information

acquired from other publications is appropriately acknowledged.

Mark Glickman

Signed:………………………………………. Date: 26 November

2007

Abstract

Power system generators are interconnected in a distributed network to allow sharing

of power. If one of the generators cannot meet the power demand, spare power is

diverted from neighbouring generators. However, this approach also allows for

propagation of electric disturbances. An oscillation arising from a disturbance at a

given generator site will affect the normal operation of neighbouring generators and

might cause them to fail. Hours of production time will be lost in the time it takes to

restart the power plant. If the disturbance is detected early, appropriate control

measures can be applied to ensure system stability. The aim of this study is to

improve existing algorithms that estimate the oscillation parameters from acquired

generator data to detect potentially dangerous power system disturbances.

When disturbances occur in power systems (due to load changes or faults), damped

oscillations (or “modes”) are created. Modes which are heavily damped die out

quickly and pose no threat to system stability. Lightly damped modes, by contrast,

die out slowly and are more problematic. Of more concern still are “negatively

damped” modes which grow exponentially with time and can ultimately cause the

power system to fail. Widespread blackouts are then possible. To avert power system

failures it is necessary to monitor the damping of the oscillating modes. This thesis

proposes a number of damping estimation algorithms for this task. If the damping is

found to be very small or even negative, then additional damping needs to be

introduced via appropriate control strategies.

This thesis presents a number of new algorithms for estimating the damping of modal

oscillations in power systems. The first of these algorithms uses multiple orthogonal

sliding windows along with least-squares techniques to estimate the modal damping.

This algorithm produces results which are superior to those of earlier sliding window

algorithms (that use only one pair of sliding windows to estimate the damping). The

second algorithm uses a different modification of the standard sliding window

damping estimation algorithm – the algorithm exploits the fact that the Signal to

Noise Ratio (SNR) within the Fourier transform of practical power system signals is

typically constant across a wide frequency range. Accordingly, damping estimates

3

are obtained at a range of frequencies and then averaged. The third algorithm applied

to power system analysis is based on optimal estimation theory. It is computationally

efficient and gives optimal accuracy, at least for modes which are well separated in

frequency.

Key words: Distributed power system, power system disturbance, power system

oscillation, white noise, coloured noise, damping factor, mean-square error,

Cramer-Rao bound, orthogonal windows, least-squares techniques, spectral

averaging, optimal estimation theory, Prony analysis.

4

List of Publications

Published conference papers and journal articles:

• Mark Glickman, Peter O’Shea, “Damping estimation of electric disturbances in

distributed power systems”, 7th IASTED International Conference on Signal and

Image Processing, Honolulu, Hawaii, USA, August 2005.

• Mark Glickman, Peter O’Shea, Gerard Ledwich, “Damping estimation in highly

interconnected power systems”, IEEE Region 10 – TENCON’05, Melbourne,

Australia, November 2005.

• Mark Glickman, Peter O’Shea, Gerard Ledwich, “Estimation of modal damping

in power networks”, IEEE Transactions on Power Systems, volume 22, issue 3,

pp. 1340 - 1350, August 2007.

• Mark Glickman and Peter O’Shea, “Optimal estimation theory based methods for

determining damping in power systems”, to be submitted to IEEE Transactions

on Power Systems in 2008.

5

Acknowledgements

I want to express my appreciation for the guidance provided my principal

supervisor Professor Peter O’Shea, my associate supervisor Professor Gerard

Ledwich and my second associate supervisor Dr. Edward Palmer.

I want to acknowledge the help of QUT academic staff and my fellow

postgraduate students. I would like to thank the QUT and Mathworks technical

support people for help with software used in this research. I also want to thank the

people that came to my seminars and conference paper presentations.

Finally I would like to thank my parents for additional assistance in this

research.

6

Table of Contents

Statement of Sources 2

Abstract 3

List of Publications 5

Acknowledgements 6

Table of Contents 7

Acronyms and Abbreviations 10

Glossary 12

List of Figures 13

List of Tables 17

1 Chapter 1: Introduction 18 1.1 Description the Problem 18

1.2 Model for “Ambient” Power System Data 19

1.3 Analysis of “Ambient” Power System Data 22

1.4 Measures of Effectiveness for Damping Estimation Methods 24

1.5 Control Measures 24

1.6 Overall Objectives of the Study 25

1.7 Specific Aims of the Study 25

1.8 Outline of the Thesis 25

1.9 References 26

2 Chapter 2: Literature Review 32 2.1 Introduction 32

2.2 Modal Parameter Estimation 33 2.2.1 Eigenanalysis Methods 33

2.2.2 Limits in Estimation Precision 33

2.2.3 Estimation Algorithms 34 2.2.3.1 Sliding Window Methods 34

2.2.3.2 High Resolution Methods 37 2.2.3.2.1 Prony Methods 39

2.2.3.2.2 High Resolution Sliding Window Method 44

2.3 Multiple Site Data Processing 45

2.4 Gaps in Knowledge 46

7

2.5 References 46

3 Chapter 3: Multiple Orthogonal Window Estimation Method 52 3.1 Introduction 52

3.2 Multiple Orthogonal Window Damping Estimation Method 55 3.2.1 Background: Basic Sliding Window Method 55

3.2.2 The Sliding Multiple Window Method 57

3.2.3 Summary of the Sliding Multiple Window Method 61

3.3 Simulations to Compare Multiple Sliding Window Method with Basic Sliding

Window Method 62

3.3.1 Additive White Noise 62

3.3.2 Additive Coloured Noise 62

3.4 High Resolution Multiple Window Method 63

3.5 Simulations 66 3.5.1 Scenario I. A Single Mode in White Noise. Damping Factor is Held Constant and

SNR is Varied 66

3.5.2 Scenario II. A Single Mode in White Noise. SNR is Held Constant and Damping

Factor is Varied 68

3.5.3 Scenario III. Two Closely Spaced Modes in White Noise. SNR is Varied. 69

3.5.4 Scenario VI. Two Heavily Damped Modes in White Noise. SNR is Varied 71

3.6 Application to a Simulated Power System Example and Real Power System

Data 74 3.6.1 Simulated Power System Example 74

3.6.2 Real Power System Example 77

3.7 Conclusion 79

3.8 References 79

4 Chapter 4: Damping Estimation Via Spectral Averaging 82

4.1 Introduction 82

4.2 Spectral Averaging Methods 84 4.2.1 Sliding Window Spectral Averaging Methods 84

4.2.2 High Resolution Sliding Window Spectral Averaging Methods 85

4.3 Simulations 87

4.4 Real Data Analysis 89

4.5 Conclusion 92

8

4.6 References 92

5 Chapter 5: Weighted Least-Squares Averaging Method for Damping

Estimation in Power Systems 95 5.1 Introduction 95

5.2 Optimal Estimation Theory Averaging Based Methods 95 5.2.1 Signal Model 95

5.2.2 Least Squares Estimators 96 5.2.2.1 Estimator 1 96

5.2.2.2 Estimator 2 98

5.2.3 Low SNR operation 99 5.2.3.1 Estimator 3 100

5.3 Simulations 101

5.4 Real Data Analysis 103

5.4.1 Processing of Power System Signals 103

5.5 Discussion 106

5.6 Conclusion 107

5.7 References 107

6 Chapter 6: General Discussion 109

6.1 Summary 109

6.2 Research Outcomes 111

6.3 Future Work 112

6.4 References 113

7 Chapter 7: Conclusion 114

8 Appendix 116

8.1 Damping Factor Estimation from Two Sliding Windows 116

8.1.1 References 118

8.2 Matlab Code used to Generate Simulated Real Data in Chapter 3 120

9

Acronyms and Abbreviations

AESOPS Analysis of Essentially Spontaneous Oscillations in Power Systems

AR Autoregressive

ARMA Autoregressive Moving Average

AWGN Additive white Gaussian noise

CRB Cramer-Rao bound

DFT Discrete Fourier Transform

DLC Daily Load Curve

DPSS Discrete Prolate Spheroidal Sequences

ERA Eigensystem Realisation Algorithm

FFT Fast Fourier Transform

FM Frequency Modulation

FT Fourier Transform

GLS Generalised Least Squares

IIR Infinite Impulse Response

HOC Higher Order Crossing

10

KT Kumaresan-Tufts

LS Least Squares

LTI Linear Time Invariant

LP Linear Prediction

MATLAB Matrix Laboratory

MMR Matrix Minimal Realisation

MSE Mean Squared Error

OMIB One Machine Interface Bus

PEALS Program for Eigenvalue Analysis of Large Systems

PDF Probability Density Function

PSS Power System Stabilisers

SIME Single Machine Equivalent

SNR Signal to Noise Ratio

SVD Singular Value Decomposition

TLS Total Least Squares

11

Glossary

Blackout – major interruption in supply of electricity.

Bracking resistor – a device that absorbs excess power.

Damping – rate of decay in oscillation magnitude.

Damping ratio – value of second-order system damping defined as.

Discrete prolate spheroidal sequences (Slepian sequences) – a group of orthogonal

windows that maximise concentration for a given bandwidth. The windows can

differ in order.

Islanding – electrical isolation of a given region from the rest of the system.

MATLAB – programming language that is suitable for mathematical simulations.

Orthogonal windows – a group of functions in which an inner product between even

and odd ordered sequences is zero.

Power system disturbance – a malfunction or fault that will affect the normal

operation of the power system.

Residual power – average power of the difference between actual and estimated time

domain signals.

Sliding window – a segment of time domain sequence.

Window – a sequence of numbers that is used to modify the discrete signal properties

via time domain multiplication.

12

List of Figures

Chapter 1: Introduction

Figure 1.1: Acquired raw data of power system disturbances 20

Figure 1.2:

Stochastic model for random power system output

component 21

Chapter 3: Multiple Orthogonal Window Estimation Method

Figure 3.1: Comparison of KT (full) and basic Prony (dashed)

damping estimate MSEs for a single mode 54

Figure 3.2:

(a) Rectangular window. (b) FT magnitude of a

rectangular window. (c) Smooth (Kaiser) window. (d) FT

magnitude of a smooth (Kaiser) window 56

Figure 3.3:

Five orthogonal windows and their FTs ( ) 4=BNw 58-59

Figure 3.4:

Comparison of Basic Sliding Window Method (dots),

Multiple Sliding Window Method (circles) and Cramer-

Rao bound (dotted line) 63

13

Figure 3.5(a): Damping MSE vs SNR for high resolution basic sliding

window (dots), high resolution multiple sliding window

(circles) and KT methods (full line), 500=KTL . The

dotted line represents the Cramer-Rao (CR) lower

variance bound 67

Figure 3.5(b):

Input sequence, noisy time domain signal 67

Figure 3.5(c):

Damping MSE vs SNR for high resolution basic sliding




variance bound 68

Figure 3.5(d):

Damping MSE vs damping for high resolution basic

sliding window (dots), high resolution multiple sliding

window (circles) and KT methods (full line), 500=KTL 69

Figure 3.5(e):

Input sequence, noiseless time domain signal 70

Figure 3.5(f):





variance bound 71

Figure 3.5(g):

Input sequence, noiseless time domain signal 72

14

Figure 3.5(h): Damping MSE vs damping for high resolution basic

sliding window (dots), high resolution multiple sliding

window (circles) and KT methods (full line), .

The dotted line represents the Cramer-Rao (CR) lower

variance bound

300=KTL

73

Figure 3.6:

Model of simulated power system 74

Figure 3.7:

Autocorrelation of data generated by the simulated power

system 76

Figure 3.8:

Fourier transform magnitude of observed signal 76

Figure 3.9:

Input sequence – time domain autocorrelation of acquired

real data 78

Figure 3.10:

Fourier transform magnitude of input sequence 78

Chapter 4: Damping Estimation via Spectral Averaging

Figure 4.1: Power system model during post disturbance oscillations 82

Figure 4.2:

High resolution sliding window spectral averaging

methods 86

Figure 4.3:

Damping MSE vs SNR for basic sliding window (dots),

average spectral estimate sliding window and average

FFT ratio sliding window (circles), average FFT sample

sliding window (squares); Kumaresan-Tufts Prony (full

line) methods 88

15

Figure 4.4:


window (dots), high resolution average spectral estimate

sliding window (circles), high resolution average FFT

sample 1 and 2 sliding window (squares); Kumaresan-

Tufts Prony (full line) methods 89

Figure 4.5:

Autocorrelation function of disturbance signal from

Tasmanian grid 90

Figure 4.6:

FFT of the autocorrelation function in Figure 4.5 90

Chapter 5: Weighted Least-Squares Averaging Method for

Damping Estimation in Power Systems

Figure 5.1: Sequence of estimates, ( )nyr for 0043.0−=α 97

Figure 5.2:

Damping MSE vs SNR for Estimator 1 (cross signs),

Estimator 2 (circles) and Prony method (plus signs).

Cramer-Rao bound is denoted by dashed line 102

Figure 5.3:

Damping MSE vs SNR for Estimator 3 (circles),

Modified Estimator 1 (cross signs) and Kumaresan-Tufts

Prony method (plus signs) for simulated single mode

data. Cramer-Rao bound is denoted by dashed line 103

Figure 5.4:

Acquired real data 105

Figure 5.5:

Fourier transform magnitude of acquired real data 105

16

List of Tables


Table 3.1: Damping estimates 77

Table 3.2:

Damping estimates MSE 77

Table 3.3:

Residual between acquired and estimated signals 79

Chapter 4: Damping Estimation via Spectral Averaging

Table 4.1: Damping estimates 91

Table 4.2:

Residual between acquired and estimated signals 91

Chapter 5: Weighted Least-Squares Averaging Method for

Damping Estimation in Power Systems

Table 5.1: Residual between acquired and estimated signals 106

17


1.1 Description of the Problem Power generators are interconnected in a distributed power network to improve the

overall quality, reliability and efficiency of power supply [1]. Interconnection

enables excess power generated at one point in the network to be diverted to another

point in the network with inadequate locally generated power. This sharing of power

eliminates the need for increasing the power output at a given generator site to meet

the local power demand. While interconnection has many advantages, it also has a

disadvantage. It allows disturbances to possibly propagate throughout the network;

that is, a disturbance in one part of the network can eventually ripple through the

entire network, and in extreme cases can cause wide-scale blackouts. Some of the

major power losses in America and Europe over the last ten years are prime

examples of the potential problems [2-12].

When disturbances do occur (eg. due to the opening and closing of circuit breakers),

or when load changes occur there is typically a shift in the rotor angle of one or more

generators [13]. This may cause a loss of synchronism with the rest of the power

system. The generator(s) in the vicinity of the disturbance/load change either

speed(s) up or slow(s) down relative to the other generators in the system. Thus the

frequency of the mains signal changes near the point of the disturbance/load change.

Then, however, the system usually tries to re-establish equilibrium and various

torsional modes (oscillations) arise, affecting the generator phase, and modulating

the mains frequency. After the modal oscillations decay the system settles at a new

operating point. If the oscillations do not settle down the generator(s) will loose

synchronism with the power system and will not be able to supply the required

amount of power to its load. Power will have to be diverted from neighboring

generators to meet the demand. If the load continues to rise, the generators may turn

off or fail, leading to blackouts.


19

When wide-scale blackouts occur, it is difficult to re-start the system and the “down-

time” contributes to significant loss of revenue. Blackouts also pose serious threats to

safety, especially for the elderly and infirmly. It is therefore important to find ways

to reduce the likelihood of system failures. One method to reduce blackout risks is to

carefully monitor power systems disturbances and instigate control measures when

potentially problematical disturbances are found. Because power system oscillations

almost invariably arise after major generator site disturbances, one simple way to

monitor for disturbances is to monitor the modal oscillations. That is, signals

obtained after major disturbances are a valuable source of information on power

system dynamics and controls [14]. The potentially dangerous scenarios are those

which have modes with light damping or even “negative” damping (the latter

corresponding to exponentially growing modes). One of the most critical tasks in

power system monitoring, then, is to rapidly estimate the damping of post-

disturbance modal oscillations. To do this estimation effectively, there is a need to

have fast and accurate damping estimation methods. This thesis concerns itself with

research into new and reliable damping estimation methods for power system

monitoring.

1.2 Model for “Ambient” Power System Data Various multiple damped oscillating modal components arise during the time it takes

a “disturbed” generator to return to its equilibrium state [15]. Two well known types

of oscillations are local and inter-area modes. Local modes are oscillations which are

largely confined to the area around a given generator; typical frequencies are 1 to 2

Hz. Inter-area modes are the oscillations between groups of machines at various

points in the system; typical frequencies are 0.2 to 0.8 Hz [16]. Unlike local modes

they tend to be lightly damped [17] (low damping) and are thus a greater threat to

system stability.

Figure 1.1 illustrates the kind of data which can be obtained after a major

disturbance. It was acquired at 1:15 am GMT, 10 April 2004 from the National


20

Australian Electricity Grid. The figure shows the voltage magnitude data acquired

from four generator sites across Australia: Adelaide Brisbane, Sydney and

Melbourne. Disturbances are apparent in the figure 4.5(a) at approximately 90

seconds and 180 seconds. While the disturbances are apparent in the figure, the

damping factors of the modes within the signal are not immediately obvious. That is

why parameter estimation algorithms, as proposed in this thesis, are required.

0 200 400

7450

7500

7550

Time (seconds)

|Vol

tage

(V)|

Adelaide

(a)

0 200 4008180

8190

8200

8210

8220

Time (seconds)

|Vol

tage

(V)|

Brisbane

(b)

0 200 400

7420

7440

7460

Time (seconds)

|Vol

tage

(V)|

Melbourne

(c)

0 200 400

7500

7550

7600

Time (seconds)

|Vol

tage

(V)|

Sydney

(d)

Figure 1.1 Acquired raw data of power system disturbances [18].

To do the parameter estimation in a practically viable way it is highly desirable that it

be done with “ambient” data. i.e. using data from power systems in normal operation,

rather than data obtained from specially devised experiments (such as braking

resistor tests or other major injected disturbances). Power systems in normal

operation intrinsically contain a large number of disturbances (such as the turning on

and off of appliances, and other events leading to variations in load). Because of this

fact, power system outputs intrinsically contain information about modal damping,


21

but this information is not encoded in a deterministic way. The ambient data from

power systems is essentially a stochastic process because it arises from a large

number of random events. Because ambient data is stochastic in nature it must be

modeled if it is to be used effectively. The subsequent paragraphs address the issue

of modeling of the power system output data.

Changes in customer load are non-deterministic, but have been found empirically to

conform reasonably well to a ‘random walk’ or Brownian noise process [19].

Accordingly, the random load change in a system in normal operation typically has a

normal distribution and a fractal frequency spectrum [20]. That is, the random load

change evolution is similar to that which would be obtained if one integrated white

noise. This random load change evolution drives the power system, which can be

reasonably well modeled as an infinite impulse response (IIR) filter [20-26]. The IIR

filter’s poles (or resonances) correspond to the modes of the system. That is, the real

parts of the IIR filter’s poles are the damping factors of the systems modes, while the

imaginary parts of the poles are the frequencies of the system modes. The random

component of the power system output ( ( )ny ), can be modeled as the output of an

IIR filter which is driven by integrated white noise. The model described above is

depicted in Figure 1.2.

Power System (IIR) filter Differentiator Integrator WGN

≡ Power system (IIR)

filter WGN

( )nx

Modal oscillation( )ny

( )nx

Figure 1.2 Stochastic model for random power system output component.

Given the stochastic model in Figure 1.2, one can estimate modal damping given

power system output measurements. In practice, one is interested in the non-


22

deterministic component of the power system output signal ( ( )ny ), which is obtained

by removing the deterministic 50 Hz mains component from the overall output. To

extract damping information from output signal, recourse can be made to various

techniques of stochastic analysis. These techniques include autocorrelation function

formation [27], probability density function (PDF) characterization [27],

thresholding [28], and minimisation of mean square errors (MSEs) [28]. These

techniques are illustrated briefly below and more fully in Chapters 3 to 5 of this

thesis.

1.3 Analysis of “Ambient” Power System Data By forming the autocorrelation of the signal, ( )nx , it is possible to obtain a signal

which contains the modal oscillation components associated with the power system

of interest, albeit with some additive noise present (See Chapter 4 for details). That

is, the autocorrelation of , has the form: ( )nx )(nzr

( ) ( ) ( )∑=

− ++=M

mmm

nmwr neAnznz m

1cos φωα , (1.1)

where amplitude, damping factor, frequency,

phase, Additive noise,

thm mA = th

m m=α thm m=ω

thm m=φ ( ) =nzw =M Number of modal oscillation

components. Frequency is assumed to be constant.

The signal in (1.1) is in a form which is readily amenable to parameter estimation.

There are various existing algorithms for estimating the amplitudes, frequencies and

damping factors [29-43]. This thesis also presents some new algorithms.

It is sometimes convenient to work with complex exponential signals (as opposed to

sinusoids) because such signals have a one-sided, rather than 2-sided spectrum. They

are therefore easier to process. One can easily obtain complex signals from real ones


23

by forming the “analytic signal” from the real one [44]. The signal model in (1.1) can

be re-stated for analytic signals as:

( ) ( ) ∑=

++−+=M

m

jnjmwr

mmmeAnznz1

)( φωα (1.2)

While damping estimation from post-disturbance records is the focus of work in this

thesis for monitoring problematical power system scenarios, it should be pointed out

that other techniques exist. Some examples of these alternative techniques are

discussed in the following paragraphs.

Drifts in frequency tend to accompany increases in system load and machines will

slow down under increased load (the mains frequency is proportional to the speed of

the generator). Detecting frequency variations can therefore be useful for identifying

major disturbances. Minor frequency variations during peak load conditions are not

considered as disturbances. A set instantaneous frequency rate threshold level must

be exceed before the frequency variation can be considered as potentially dangerous

disturbance that requires appropriate control measures. Examples of this are provided

in [23].

Problematical disturbances can also be detected and classified with wavelet

transforms [45]. Recognition can be done in both the time and frequency domains.

Wavelets allow short computational times but must be chosen correctly, otherwise

results can be inaccurate or even wrong [45]. Identifying closely spaced modes is

particularly challenging. Kam [46] used wavelets to classify time domain circuit

beaker re-strike records. Specific waveforms were identified with neural networks;

square wavelet transform coefficients were fed as inputs into the neural network. The

disturbance type can be also identified by combining the neural networks outcomes

via various decision making schemes.


24

1.4 Measures of Effectiveness for Damping Estimation

Methods To evaluate new methods of damping estimation it is necessary to have a

performance metric. If the damping is known (as it is with simulated data) then one

can use the mean squared error (MSE) between the true and estimated damping. If

the damping is not known (at is the case with real data) then one can use the mean

squared difference between the actual waveform and the waveform reconstructed

from the estimated signal parameters. These two different metrics are used

throughout the thesis for evaluating performance on simulated and real data.

1.5 Control Measures Once modal damping has been estimated it is necessary to determine whether

explicit control measures need to be used. Such control measures can be used to

attenuate the modal oscillations. One of those measures involves the use of power

system stabilizers (PSS). The analysis of modal parameters is important information

that is used for adjustment of these devices [31].

Power system stability is also maintained via various intelligent load shedding

procedures [47-49]. Once the location of a disturbance is found the loads at the

generator site are shed till the generator frequency stops dropping [50]. A load

restoration procedure is initiated when the generator frequency exceeds 50 Hz [50].

After load restoration the generator frequency settles (hopefully) to a steady value of

50 Hz.

Power load curves tend to show a spike at a point when the load starts dropping [50].

In an event of a disturbance, certain regions can be isolated from the main network.

This separation, also known as islanding, involves supply of power from local

generators and shedding of load [48, 50].


25

1.6 Overall Objectives of the Study The chief objective of this research is to improve the quality of damping estimation

for electromechanical modes in power systems. A second aim is devise algorithms

which can achieve the improved damping estimation rapidly, thereby enabling any

necessary remedial strategies in a timely manner.

1.7 Specific Aims of the Study 1. Improve existing sliding window damping factor estimation techniques via

various averaging methods.

2. Investigate the affect of windows properties on damping mean square error.

3. Compare new techniques with Prony methods.

4. Test new algorithms on real data.

5. Test new algorithms on multiple site data acquired from sites across Australia

and use averaging techniques to find the average damping for the interconnected

system.

1.8 Outline of the Thesis Chapter 2 of this thesis presents a literature review of the damping estimation

methods which are currently used in power system monitoring. Chapter 3 describes

and analyses new estimation methods based on the use of orthogonal sliding

windows. The work presented in this chapter was published in the IEEE

Transactions on Power Systems, 2007. Chapter 4 presents another new technique

based on a combination of sliding window and spectral averaging techniques. Some

of the work presented in this chapter has been published in the Proceedings of IEEE

TENCON, 2005. Chapter 5 applies techniques in optimal estimation to power

systems for the first time. The work in this chapter is to be submitted to the IEEE

Transactions on Power Systems. Chapter 6 is devoted to a discussion of the results

and to possible future areas of investigation. Conclusions are provided in Chapter 7.


26

Finally, Chapter 8 contains some statistical analysis for the work described in

Chapter 3 and Matlab code.

1.9 References [1] M. Aktarujjaman, M. A. Kashem, M. Negnevitsky, and G. Ledwich,

"Dynamics of a hydro-wind hybrid isolated power system," Proc. AUPEC

Conf., Sep. 2005.

[2] J. E. Dagle, "Data management issues associated with the August 14, 2003

blackout investigation," Proc. IEEE Power Eng. Soc. Gen. Meet., vol. 2, pp.

1680 - 1684, Jun. 2004.

[3] G. Andersson, P. Donalek, R. Farmer, N. Hatziargyriou, I. Kamwa, P.

Kundur, N. Martins, J. Paserba, P. Pourbeik, J. Sanchez-Gasca, R. Schulz, A.

Stankovic, C. Taylor, and V. Vittal, "Causes of the 2003 major grid blackouts

in North America and Europe, and recommended means to improve system

dynamic performance," IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1922 -

1928, Nov. 2005.

[4] G. I. Maldonado, "The performance of North American nuclear power plants

during the electric power blackout of August 14, 2003," IEEE Nuclear

Science Symposium Conf. Record, vol. 7, pp. 4603 - 4606, Oct. 2004.

[5] B. Yang, V. Vittal, and G. T. Heydt, "Slow-Coherency-Based Controlled

Islanding - A Demonstration of the Approach on the August 14, 2003

Blackout Scenario," IEEE Trans. Power Syst., vol. 21, no. 4, pp. 1840 - 1847,

Nov. 2006.

[6] S. Corsi and C. Sabelli, "General blackout in Italy Sunday September 28,

2003, h. 03:28:00," IEEE Power Eng. Soc. Gen. Meet., vol. 2, pp. 1691 -

1702, Jun. 2004.

[7] A. Berizzi, "The Italian 2003 blackout," IEEE Power Eng. Soc. Gen. Meet.,

vol. 2, pp. 1673 - 1679, Jun. 2004.


27

[8] Y. Hain and I. Schweitzer, "Analysis of the power blackout of June 8, 1995 in

the Israel Electric Corporation," IEEE Trans. Power Syst., vol. 12, no. 4, pp.

1752 - 1758, Nov. 1997.

[9] J. F. Hauer, N. B. Bhatt, K. Shah, and S. Kolluri, "Performance of "WAMS

East" in providing dynamic information for the North East blackout of

August 14, 2003," Proc. IEEE Power Eng. Soc. Gen. Meet., vol. 2, pp. 1685 -

1690, Jun. 2004.

[10] C. W. Taylor and D. C. Erickson, "Recording and analyzing the July 2

cascading outage [Western USA power system]," IEEE Computer Appl. in

Power, vol. 10, no. 1, pp. 26 - 30, Jan. 1997.

[11] S. Lindahl, G. Runvik, and G. Stranne, "Operational experience of load

shedding and new requirements on frequency relays," 6th Int. Conf.

Developments in Power Syst. Protect., pp. 262 - 265, Mar. 1997.

[12] S. Larsson and E. Ek, "The black-out in southern Sweden and eastern

Denmark, September 23, 2003," IEEE Power Eng. Soc. Gen. Meet., vol. 2,

pp. 1668 - 1672, Jun. 2004.

[13] G. Ledwich, "Identification of high power loads," Proc. AUPEC Conf., Sep.

2004.

[14] J. Hauer, D. Trudnowski, G. Rogers, B. Mittelstadt, W. Litzenberger, and J.

Johnson, "Keeping an eye on power system dynamics," IEEE Computer Appl.

in Power, vol. 10, no. 4, pp. 50 - 54, Oct. 1997.

[15] P. O'Shea, "An algorithm for power system disturbance monitoring," Proc.

IEEE Int. Conf. Acoust., Speech, and Sig. Proc. (ICASSP), vol. 6, pp. 3570 -

3573, Jun. 2000.

[16] T. George, J. Crisp, and G. Ledwich, "Advanced tools to manage power

system stability in the national electricity market," Proc. AUPEC Conf., Sep.

2004.

[17] E. W. Palmer and G. Ledwich, "Optimal placement of angle transducers in

power systems," IEEE Trans. Power Syst., vol. 11, no. 2, pp. 788 - 793, May

1996.


28

[18] G. Ledwich and C. Zhang, "Disturbance Report (unpublished work),"

Unpublished, Queensland University of Technology, Brisbane, Australia,

Apr. 2004.

[19] G. Ledwich and E. Palmer, "Modal estimates from normal operation of power

systems," Proc. IEEE Power Eng. Soc. Wint. Meet., vol. 2, pp. 1527 - 1531,

Jan. 2000.

[20] R. A. Wiltshire, P. O`Shea, and G. Ledwich, "Rapid detection of deteriorating

modal damping in power systems," Proc. AUPEC Conf., Sep. 2004.

[21] R. A. Wiltshire, "The analysis of disturbance modes in large interconnected

power systems," PhD Confirmation of Candidature Report, Queensland

University of Technology, Brisbane, Australia, 2003.

[22] R. A. Wiltshire, "Summary of PhD confirmation of candidature report: The

analysis of disturbance modes in large interconnected power systems,"

Queensland University of Technology, Brisbane, Australia, 2003.

[23] R. A. Wiltshire, "Analysis of disturbances in large interconnected power

systems," PhD Thesis, Queensland University of Technology, Brisbane,

Australia, 2007.

[24] R. A. Wiltshire, P. O`Shea, and G. Ledwich, "Rapid detection of changes to

individual modes in multimodal power systems," IEEE Region 10 - TENCON

'05, Nov. 2005.

[25] R. A. Wiltshire, P. O'Shea, and G. Ledwich, "Monitoring of individual modal

damping changes in multi-modal power systems," Australian Journ. of

Electrical and Electronic Eng., vol. 2, no. 3, Jan. 2006.

[26] R. A. Wiltshire, P. O'Shea, G. Ledwich, and M. Farquharson, "Application of

statistical characterisation to the rapid detection of deteriorating modal

damping in power systems," 7th IASTED Int. Conf. Signal and Image Proc.,

Aug. 2005.

[27] A. Papoulis and S. U. Pillai, Probability, random variables, and stochastic

processes, 4th ed. Boston: McGraw-Hill, 2002.

[28] S. M. Kay, Fundamentals of statistical signal processing: estimation theory,

Englewood Cliffs, N.J.: PTR Prentice Hall, 1993.


29

[29] J.-H. Hong and J.-K. Park, "A time-domain approach to transmission network

equivalents via Prony analysis for electromagnetic transients analysis," IEEE

Trans. Power Syst., vol. 10, no. 4, pp. 1789 - 1797, Nov. 1995.

[30] M. A. Johnson, I. P. Zarafonitis, and M. Calligaris, "Prony analysis and

power system stability-some recent theoretical and applications research,"

Proc. IEEE Power Eng. Soc. Summer Meet., vol. 3, pp. 1918 - 1923, Jul.

2000.

[31] M. Amono, M. Watanabe, and M. Banjo, "Self-testing and self-tuning of

power system stabilizers using Prony analysis," Proc. IEEE Power Eng. Soc.

Wint. Meet., vol. 1, pp. 655 - 660, Jan. - Feb. 1999.

[32] M. Meunier and F. Brouaye, "Fourier transform, wavelets, Prony analysis:

tools for harmonics and quality of power," Proc. IEEE 8th Int. Conf.

Harmonics And Quality of Power, vol. 1, pp. 71 - 76, Oct. 1998.

[33] P. S. Dolan, J. R. Smith, and W. A. Mittelstadt, "Prony analysis and modeling

of a TCSC under modulation control," Proc. 4th IEEE Int. Conf. Control

Appl., pp. 239 - 245, Sep. 1995.

[34] P. K. Gale and J. W. Pierre, "Prony analysis based parameter estimation of an

NMR signal of blood plasma for cancer detection," Proc. IEEE Int. Conf.

Acoust., Speech, and Sig. Proc. (ICASSP), vol. 2, pp. 1185 - 1188, May 1995.

[35] J. C. Mosher and P. S. Lewis, "Taylor series expansion and modified

extended Prony analysis for localization," Proc. IEEE 28th Asilomar Conf.

Sig., Syst. and Computers, vol. 1, pp. 667 - 670, Oct. - Nov. 1994.

[36] E. Palmer, "The use of Prony analysis to determine the parameters of large

power system oscillations," Proc. AUPEC Conf., Sep. - Oct. 2002.

[37] C. E. Grund, J. J. Paserba, J. F. Hauer, and S. L. Nilsson, "Comparison of

Prony and eigenanalysis for power system control design," IEEE Trans.

Power Syst., vol. 8, no. 3, pp. 964 - 971, Aug. 1993.

[38] J. F. Hauer, C. J. Demeure, and L. L. Scharf, "Initial results in Prony analysis

of power system response signals," IEEE Trans. Power Syst., vol. 5, no. 1,

pp. 80 - 89, Feb. 1990.


30

[39] J. F. Hauer, "Application of Prony analysis to the determination of modal

content and equivalent models for measured power system response," IEEE

Trans. Power Syst., vol. 6, no. 3, pp. 1062 - 1068, Aug. 1991.

[40] D. J. Trudnowski, J. M. Johnson, and J. F. Hauer, "Making Prony analysis

more accurate using multiple signals," IEEE Trans. Power Syst., vol. 14, no.

1, pp. 226 - 231, Feb. 1999.

[41] D. A. Pierre, D. J. Trudnowski, and J. F. Hauer, "Identifying linear reduced-

order models for systems with arbitrary initial conditions using Prony signal

analysis," IEEE Trans. Automatic Control, vol. 37, no. 6, pp. 831 - 835, Jun.

1992.

[42] D. J. Trudnowski, J. M. Johnson, and J. F. Hauer, "SIMO system

identification from measured ringdowns," Proc. American Control Conf., vol.

5, pp. 2968 - 2972, Jun. 1998.

[43] A. A. Beex and P. Shan, "A time-varying Prony method for instantaneous

frequency estimation at low SNR," Proc. IEEE Int. Symposium Circuits and

Syst. (ISCAS), vol. 3, pp. 5 - 8, May - Jun. 1999.

[44] R. N. Bracewell, The Fourier transform and its applications, 2nd ed., rev. ed.

New York: McGraw-Hill, 1986.

[45] X. Dong-jie, H. Ren-mu, and X. Tao, "A new approach to power system

electrocmechanical oscillation research," Power Syst. and Communs.

Infrastructures for the Future, 2002.

[46] S.-C. Kam and D. Birtwhistle, "Monitoring high voltage circuit breaker

condition using wavelet transform and artificial intelligent modelling

(unpublished work)," Unpublished, Queensland University of Technology,

Brisbane, Australia, 2004.

[47] Y.-W. Au and K.-M. Yung, "Emergency control and restoration of power

system under disturbance," Proc. IEEE Int. Conf. Advances in Power Syst.

Control, Operation and Management (APSCOM), vol. 2, pp. 808 - 812, Nov.

1997.


31

[48] S. S. Ahmed, N. C. Sarker, A. B. Khairuddin, M. R. B. A. Ghani, and H.

Ahmad, "A scheme for controlled islanding to prevent subsequent blackout,"

IEEE Trans. Power Syst., vol. 18, no. 1, pp. 136 - 143, Feb. 2003.

[49] J. Jung, C.-C. Liu, S. L. Tanimoto, and V. Vittal, "Adaptation in load

shedding under vulnerable operating conditions," IEEE Trans. Power Syst.,

vol. 17, no. 4, pp. 1199 - 1205, Nov. 2002.

[50] M. Banejad, G. Ledwich, and M. A. Kashem, "Operation of power system

islands," Proc. AUPEC Conf., Sep. 2005.

Chapter 2: Literature Review

2.1 Introduction As discussed in the previous chapter, disturbances can occur in power systems due to

faults or load changes, and when such disturbances do occur modal oscillations

usually arise. It is important to monitor the damping of these oscillations to ensure

that there are no modes which are too lightly (or even negatively) damped. There are

a number of practical techniques which can be used to estimate the damping of the

modes from power system outputs. One way is to deliberately cause a sudden change

to the power system (with say a “braking resistor test”) and then use parameter

estimation for the modal oscillations in the ensuing output. This has the disadvantage

of requiring a major disturbance to the power system. The advantage is that a high

signal to noise ratio (SNR) output signal is usually obtained and the parameter

estimation can be performed quite quickly. A second way to obtain damping

estimates is to use the output from power systems in normal operation (i.e. using

ambient data) and infer damping information with stochastic modeling techniques.

As will be explained in more detail in subsequent chapters, this process involves a

pre-processing stage, an autocorrelation function formation stage and finally a

parameter estimation stage. This second approach has the advantage that the power

system does not have to be unnecessarily disturbed. It has the disadvantage that

much longer times are required to be able to estimate damping accurately.

Both the braking resistor test method and the ambient data method require the use of

damping estimation from a time-domain signal. The following section describes the

existing parameter estimation methods for such time domain estimation.


33

2.2 Modal Parameter Estimation 2.2.1 Eigenvalue Analysis Methods Traditionally, modal estimation has been performed in power systems with off-line

analytical methods. That is, machine inertias, line impedances and component

models are developed to predict the modes that would arise when a particular power

system is disturbed. Once the model is created, the modal resonances of the system

can be determined (assuming small signal perturbations) by eigen analysis methods

[1-12] (including Analysis of Essentially Spontaneous Oscillations in Power Systems

(AEPSOPS) and Program for Eigenvalue Analysis of Large Systems (PEALS) [1]).

One of the difficulties with this approach is that the ability to predict the modal

parameters is critically dependant on the validity of the assumed model. As a

consequence, estimation of modal parameters from power system output

measurements has become much more popular in recent times [2-25]. This latter

approach is the basis of the work done in this thesis. A review of the methods

currently available to do this estimation is provided below.

2.2.2 Limits in Estimation Precision For a given signal model and associated noise model there is a fundamental limit to

how accurately the model parameters can be estimated. This limit is known as the

Cramer-Rao bound (CRB). The CRBs represents the minimum mean square errors

(MSEs) for a set of unbiased parameter estimates [26]. The process for determining

the CRBs is straightforward in principle, but can frequently lead to complex (and

even intractable) mathematical derivations. The CRBs are the diagonals of the

inverse of the Fisher information matrix [25], the latter being the second derivative

of the likelihood of the observed signal, given the assumed signal and noise model.

The differentiation is with respect to the modal parameters. Cramer-Rao bound

expressions for modal parameter estimation of single and two modal components in

white Gaussian noise are derived in [25].


34

2.2.3 Estimation Algorithms In general, the quality of an algorithm is judged by its ability to yield parameter

estimates which have low MSE. When judging the performance of algorithms on real

data, however, one cannot use the MSE because one does not know the true value of

the parameter(s). For real data analysis, therefore, it is more appropriate to use

residual power as a measure of performance. The residual power is the average of the

square of the difference between the observed signal and the signal which is

reconstructed from the estimated parameters:

Residual Power ( ) ( )∑−

=

−≈1

0

2ˆ1 N

nsr nznz

N (2.1)

where Number of samples in a signal, =N ( )nzsˆ is the signal reconstructed from

modal parameter estimates and is the observed signal. ( )nzr

There are many estimation algorithms for determining the damping of exponentially

damped sinusoids. Some of these many algorithms are presented in the following

subsections.

2.2.3.1 Sliding Window Methods

The original sliding window method was presented in a paper by Poon and Lee [4].

A subsequent modification of the method was presented by the same authors in [5].

In this method two consecutive segments of the observed signal are Fourier

transformed and the relative Fourier amplitudes in these two segments (at the

frequency of the oscillating modal component) are used to evaluate the damping

factor. For example, if the mode is decaying rapidly, the Fourier amplitude in the

second window will tend to be significantly less than in the first window.

The damping estimation method proposed in [4, 5] had a major limitation. It could

only be implemented with certain restricted window lengths. O’Shea later noted that

this restriction was due to the interference between “positive” and “negative”


35

frequency components [15] (Poon and Lee [4] had specified discrete values of

window length at which interference between those frequency components was zero

( 0/ωπmNw = , where ∞= K,3,2,1,0m ).). O’Shea also showed that the restriction

could be removed if smoothly tapering windows were used (so that this interference

was strongly mitigated) rather than conventional rectangular ones. Providing

smoothly tapering windows are used damping of well separated modes can be readily

estimated by appropriately scaling the logarithm of the ratio of the two Fourier

amplitudes [26]:

( ) ( )

( ) ( ) ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+=

∑

∑−

=

−

−

=

−

1

0

1

0

0

0

log1ˆw

w

N

n

njgr

N

n

njr

g eNnznw

enznw

N ω

ω

α (2.2)

where =α Modal damping estimate, ( ) =nzr Observed signal, Window, ( ) =nw

=0ω Modal frequency estimate, =n discrete time, =gN Number of samples between

windows and Number of samples in a window. =wN

Often the frequency of the mode will not be known a priori, but can be estimated

quite accurately from the Fourier transform of the observed signal. The frequency

estimate will be denoted by 0ω .

The amplitude and phase of the mode can also be readily estimated with the


36

following estimators:

( ) ( )

( )∑

∑−

=

−

−

=

−

= 1

0

1

0ˆ

0

ˆw

w

N

n

n

N

n

njr

j

enw

enznweA

α

ω

φ (2.3)

The finding in [15] that there was no restriction on the window length for damping

estimation paved the way for “optimisation” of the window length. The optimisation

was done so as to minimise the MSE of the damping estimate, assuming additive

white Gaussian noise (AWGN). The formula for the optimal window length is

derived in [15].

The basic sliding window methods in [4, 15] can be modified and extended to

improve the accuracy of the damping estimates. One such extension is proposed in

Chapter 3 of this thesis. It involves using multiple orthogonal windows in

conjunction with least-squares averaging.

When multiple modal components are present and are well separated in the

frequency domain, one can use filtering to reduce influence of the neighbouring

modal components on the sliding window parameter estimates. An alternative and

faster procedure involves determining the parameters of the mode and then removing

estimated components from the observation via subtraction. Higher energy

components are removed before lower energy components.

Sliding window methods are not reliable for parameter estimation of closely spaced

frequencies due to interferences between spectral magnitudes. Large spectral peaks

will shift the maximums of smaller spectral peaks. Thus higher damped modes can

be hidden under lower damped modes [24]. O’Shea [13] stated that sliding window

methods cannot estimate parameters of modal components closer than apart.

The frequencies should ideally be separated by more than

(NT/1 )( )TNw/4 or ( )TNw/8 ,


37

depending on the type of window used [13]. In selecting a window to minimize

interference one needs low sidelobes and a narrow mainlobe. The Kaiser window is

quite useful in trying to achieve both of these goals because one of its parameters

(“beta”) allows the user to make the trade-off between sidelobe height and main-lobe

width.

Poon and Lee argued that resolution of closely spaced modal components could be

improved by using only the imaginary components of Fourier transforms [5]. Their

claim is questionable, however, since their examples appeared to assume knowledge

of some parameters of one of the components. This knowledge, however, is not

usually available a priori.

2.2.3.2 High Resolution Methods

Because of the difficulties in using conventional Fourier methods to analyse closely

spaced and “hidden” modal components, parametric estimation methods are often

used. Note that ‘hidden’ modes are ones that are highly damped and are not apparent

in the Fourier spectrum due to interference from lightly damped modes. The

parametric approach involves modelling the disturbance as the impulse response of a

linear time invariant system. The damping factor(s) can then be extracted from the

system pole(s), also known as the eigenvalues.

Autoregressive Moving Average (ARMA) parameter estimation methods [27, 28, 29,

30] model the observation as the output of a system driven by white noise. The

system poles contain the damping and frequency information. The system discrete

time model for the system transfer function is [27]:

( ) ( )( ) n

nnnm

mmm

bzbzbzazazaz

zBzAzH

++++++++

== −−

−−

K

K2

21

1

22

11 (2.4)

The ARMA parameter estimation process consists of two parts, numerator

polynomial based parameter estimation and denominator polynomial based


38

parameter estimation. The denominator polynomial is typically more important than

the numerator because it contains the characteristic equation from which poles can be

extracted. The latter contain the vitally important damping factor information.

The denominator polynomial can be found by solving linear prediction equations

which are also known as the modified or extended Yule-Walker equations [31, 32,

33]. The denominator parameters are related linearly to the autocorrelation function,

[27]: kp

02211 =++++ −−− nknkkk pbpbpbp K for (2.5) mk >

where:

( )( ) ( )( )

( )( )∑

∑

=

−

=

−

−−−= 2/

1

2

2/

1N

nr

kN

nrr

k

nz

nzknzp

μ

μμ 2/2/ NkN <<− (2.6)

and

( )⎟⎠

⎞⎜⎝

⎛= ∑

=

2/

1

2 N

nny

Nμ (2.7)

The denominator parameters can be estimated from the autocorrelation function

values by solving a set of simultaneous linear equations. This is known as the

modified or extended Yule-Walker method where it is used to estimate

autoregressive (AR) models in the presence of noise:

pbΩ ˆˆˆ −= (2.8)

where:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

−+

+

1

1

ˆ

ˆˆ

ˆ

nk

k

k

p

pp

Mp , and

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−=

−+

+

1

1

ˆ

ˆˆ

ˆ

nk

k

k

b

bb

Mb

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

−−+−+

+−−

−−−

132

11

21

ˆˆˆ

ˆˆˆˆˆˆ

ˆ

knknk

nkkk

nkkk

ppp

pppppp

L

MOMM

L

L

Ω


39

The poles are estimated by finding the roots of: ( ) nnnn bzbzbzzB ++++= −− K2

21

1 .

ARMA estimation via the use of the autocorrelation function requires data storage

for the values. The use of higher order crossing can eliminate the need for data

storage [27]. The autocorrelation lag values are calculated from normalised Higher-

Order Crossing (HOC) sequence

kp

nmDDD +ˆ,,ˆ,ˆ

21 K , formed by counting the number

of times the observed signal, ( )nzr crosses its expected value.

2.2.3.2.1 Prony Methods

Prony methods [21, 23, 25, 34-46] assume that the system has a constant numerator

and a polynomial denominator transfer function:

( ) LL zbzbzbb

GzH −−− ++++=

K22

110

(2.9)

where L is the polynomial order. The denominator polynomial is also known as the

prediction error filter polynomial. Its coefficients are found by solving the linear

equation in a least-square sense:

( ) ( ) ( )∑=

−=L

irr inzibnz

1

(2.10)

That is, traditional Prony methods find the filter polynomial coefficients by

minimising:

( ) ( ) ( )2

1∑ ∑= =

−−=N

Ln

L

krr knzkbnzE (2.11)


40

The equation is arranged into a linear prediction matrix [36]:

vAb = (2.12)

where:

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−

−−−

=

132

11021

LNzNzNz

zLzLzzLzLz

rrr

rrr

rrr

L

MOMM

L

L

A

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

Lb

bb

M2

1

, b and

( )( )

( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

+=

1

1

Nz

LzLz

r

r

r

Mv

Alternatively a backward direction linear prediction formulation can be used [25]:

vAb −= (2.13)

where:

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−+−−

+=

11

13221

***

***

***

NzLNzLNz

LzzzLzzz

rrr

rrr

rrr

L

MOMM

L

L

A

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

Lb

bb

M2

1

, b and

( )( )

( )⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−−

=

1

10

*

*

*

LNz

zz

r

r

r

Mv

The linear prediction vector, b , in (2.12-2.13) can readily be solved using least-

squares inversion. The problem with traditional Prony estimators is that the least-

squares solution is often ill-conditioned. Kumaresan and Tufts [25] proposed that

Singular Value Decomposition (SVD) be used to improve the conditioning of the

least-squares inversion process. This improvement is achieved by first finding the

SVD of A . i.e. by finding the following factors for A :

TUSVA = (2.14)

where contains the eigenvectors of , contains the eigenvectors of

and is a diagonal matrix containing the eigenvalues. To reduce the effects of poor

U AAT V TAA

S


41

conditioning and noise, the rank of is reduced. That is, the components

corresponding to very small eigen values (and the corresponding eigen vectors) are

assumed to be due to noise rather than signal and are discarded. The conditioning

parameter is useful in determining what singular values should be discarded; this

parameter is the ratio of the maximum to minimum singular value magnitudes. In

practice the rank of

A

A should ideally be made equal to twice the number of modes

present, assuming a real signal [25]. In practice one often does not know what ill-

conditioning parameter is determined by ratios of maximum to minimum singular

value magnitudes.

Simulations in [25] showed that using the reduced rank approximation (described

above) in the inversion improves the accuracy significantly. Simulations in [25] also

showed that MSE of poles estimates can be reduced by increasing the linear

prediction model order, L [23, 47]. Hauer [43] stated that higher order models

should be used in power system scenarios where there are severe noise conditions,

although one does then run the risk that the algorithm fits modal components to the

noise [46]. There is also a trade-off when using large values of L . Although the

accuracy is generally higher, the computation is also more intense.

Several recommendations are made in [25] to promote accurate and robust parameter

estimation. One of these recommendations is that backward prediction be used to set

up and solve the linear prediction equations. If this is done then the estimated filter

transfer function has roots which are on the opposite side of the unit circle to the true

roots. It is therefore relatively easy to identify the true and extraneous roots. Another

recommendation is that (for heavily damped modes in particular) “radial bias”

correction be performed. This bias is common in linear prediction (LP) methods

when substantial noise is present. In the Kumaresan and Tufts method the bias

manifest as additional magnitude on all the singular values. The bias can be reduced

by subtracting the average of the extraneous singular values corresponding from the

true (signal) singular values. This technique is effective for single and multiple

modes.


42

In Prony type methods, robustness to noise can also be enhanced with interleaved

methods (Ledwich and Palmer [48]). Instead of 1-step prediction, k-step prediction is

used to set up and solve the linear prediction equations. This approach is an

alternative to reducing the sampling rate, which has a similar effect. Similarly to

sampling rate reduction upper limit on is imposed by aliasing [48]. The linear

prediction model equation for interleaved Prony is:

k

vAb = (2.15)

where:

( )( ) ( )( ) ( )( )( ) ( )( ) ( )

( )( ) ( )( ) ( )( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−

−−−

=

132

1021

LNkzNkzNkz

kzLkzLkzzLkzLkz

rrr

rrr

rrr

L

MOMM

L

L

A

( )( )

( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

Lb

bb

M

21

, b

and

( )( )( )

( )( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

+=

1

1

Nkz

LkzkLz

r

r

r

Mv

Trudnowski [23] showed that Prony’s method can be readily extended to data from

multiple sites. One simply augments the linear prediction equation set to include the

data from the different sites. Least-squares inversion of this equation set then

proceeds as usual.

There are a number of methods similar to conventional Prony analysis which can be

used for parameter estimation of closely spaced modes. One of these is the

eigensystem realisation algorithm (ERA) [49]. Similarly to Prony method this

method can be applied to estimation of closely spaced modes. SVD of a Hankel

matrix, is computed, such that , where and are singular vectors

and is a diagonal matrix. The Hankel matrix, , is analogous to Prony’s linear

0H T0 PSQH = P Q

S 0H


43

prediction matrix. Selected singular values and vectors in , and Q are used to

construct the state matrix from which system input and output matrices are found.

Those matrices can be used to determine the eigenvalues. Other variants of Prony

analysis include the total least squares (TLS) method presented in [47] and the

generalised least squares (GLS) method [50]. In the former there is a scheme for

simultaneously reduction of the affect of noise on the linear prediction matrix and the

observation vector. In the latter the system transfer function is estimated from time

sequence data, and curve fitting of various models is used. The model order is

increased to obtain a good fit between modelled and acquired data. The advantage of

GLS over standard LS is the ability to detect low frequency modes via appropriate

filtering of the observed signal.

P S

Yet another variant of the Prony method which was used specifically for the

purposes of estimating the parameters of “hidden modes” was the so called “T-

matrix method” presented in [24]. Recall that “hidden” modes are highly damped and

are not apparent in the Fourier spectrum due to interference from lightly damped

modes. As explained in [24] the T-matrix is computed and is used to extract the

modes from acquired voltage angles of real power system data. The autocorrelation

function of the resulting sequence is applied to obtain the system response which is

then fed into a least squares fit algorithm to determine the system poles. Estimated

modes can be used for T-matrix refinement iteration to improve the modal

separation. The method was successfully implemented on real data to estimate modal

parameters from multiple sites in [24]. Mean of estimate bias and standard deviation

were also determined.

Prony type methods are not as robust to noise as sliding window methods and are

strongly affected by non-linearities [25, 48] arising from large scale disturbances,

and the presence of noise [21]. Such problems cause significant ill-conditioning of

the linear prediction matrix and gives rise to increased MSE in parameter estimates.

These issues were explored in real power system analysis scenarios outlined in [48]

and [43].


44

)

2.2.3.2.2 High Resolution Sliding Window Method

Fourier methods are well known to be quite robust to noise. Sliding window

methods, which are Fourier based, have the advantage of robustness. They have the

disadvantage, though, that they cannot analyse closely spaced modes. The high

resolution sliding window method is a combination of sliding window algorithm and

Kumaresan-Tufts Prony method. It was introduced in [17] and has the advantage of

being able to process closely spaced modes and of being able to do this processing at

relatively low SNR thresholds [14]. The method extends the sliding window

algorithm by computing FFTs in more than two different windows, and evaluating

these FFTs at the dominant frequency in the frequency band of interest:

( ) ( ) ( ) ggg

N

m

mjri NNNNnenmzmwnF

wi −=+= ∑

−

=

− ,,2,,0ˆ,1

0

ˆ0

0 Kωω (2.16)

where:

( ) ( )∑−

=

−=1

00 maxargˆ

N

n

njri enznw ω

ωω

The resulting sequence, ( inF 0ˆ,ω , is equivalent to convolution of the observed signal

with a band-pass filter impulse response, resulting in an enhanced SNR. Because the

new sequence, ( inF 0ˆ, )ω , is a band-limited signal, it can be down-converted to

baseband and then its sampling rate can be reduced without any loss of information

[17]. The reduced sample rate enables subsequent processing to be done very

efficiently, and this can be an important issue for parameter estimation of closely

spaced modes.

Once the sequence has been down-converted to baseband and the sampling rate is

reduced one can apply the Kumaresan-Tufts method to estimating the modal

parameters. The relationship between the parameters in the original observation and

the down-shifted/down-sampled signal is discussed in [17]. Because the noise has

been reduced significantly in the band-pass filtering afforded by the sliding widow


45

method, the order of the linear prediction polynomial used in the Kumaresan-Tufts

method for the down-shifted/down-sampled signal does not need to be very large.

Simulations in [17] have shown the sliding window method to be very accurate. In

the absence of noise, precision to six decimal places was demonstrated. The amount

that the sample rate can be reduced is dictated by the Nyquist criterion. To ensure

that the sample rate reduction can be as large as possible, one should make the band-

pass filtering as narrowband as possible. This in turn implies a window with a narrow

spectral bandwidth. This further implies that the window should have negligible

energy in its sidelobes. A smooth window such as the Kaiser window is a suitable

choice [17].

2.3 Multiple Site Data Processing Data from multiple sites can be combined to obtain the overall damping estimate and

detect a post disturbance oscillation within the power system. One of the most

straightforward ways to do this is to adapt Prony’s method by concatenating the

prediction equations from the different sites into one large prediction matrix and then

solve the matrix equation in a least-squares sense as per usual. This approach was

used in [23]. Other authors have proposed different methods for doing the estimation.

For example, in [51] transients from a group of generators was processed with

multiple channel Prony method [51].

Other authors noted that the challenge was selecting the appropriate generator swing

curves out of hundreds of generators [52]. One of the proposed solutions was the

Single Machine Equivalent (SIME) method with the use of One Machine Interface

Bus (OMIB) transformation [53]. In a multi-machine power system large scale modes

could be oscillating against two “critical” machines. SIME identifies the critical

machines to create an OMIB equivalent curve that is fed into Prony algorithm. The

algorithm was applied to local and inter-area modes in [53].


46

2.4 Gaps in Knowledge There has been much which has been published on modal estimation, but there is

significant room for improvement in these methods. There is very little in the

literature about how accurate existing methods are compared to the theoretically

achievable limits. Moreover, the limited research into this area reveals that the

existing power system parameter estimation methods all appear to be sub-optimal

[17]. Chapters 3 to 5 present new methods which have improved statistical

performance when compared with existing techniques, at least for some power

system scenarios.

There is also very little consideration in the literature given to explicitly taking

account of the colour of the noise in estimation methods. This is an important

omission because the colour of the noise has a significant bearing on estimation

performance [54]. Chapters 4 and 5 address this issue explicitly.

2.5 References [1] P. W. Sauer, C. Rajagopalan, and M. A. Pai, "An explanation and

generalization of the AESOPS and PEALS algorithms [power system

models]," IEEE Trans. Power Syst., vol. 6, no. 1, pp. 293 - 299, Feb. 1991.

[2] S. Muretic, "Power system modal damping estimation," Ph.D. dissertation,

RMIT University, Melbourne, Australia, 2003.

[3] S. L. Marple, Digital spectral analysis: with applications. Englewood Cliffs,

N.J.: Prentice-Hall, 1987.

[4] K. K.-P. Poon and K.-C. Lee, "Analysis of transient stability swings in large

interconnected power systems by Fourier transformation," IEEE Trans.

Power Syst., vol. 3, no. 4, pp. 1573 - 1581, Nov. 1988.

[5] K. C. Lee and K. P. Poon, "Analysis of power system dynamic oscillations

with heat phenomenon by Fourier transformation," IEEE Trans. Power Syst.,

vol. 5, no. 1, pp. 148 - 153, Feb. 1990.


47

[6] V. Valeau, J.-C. Valiere, and C. Mellet, "Instantaneous frequency tracking of

a sinusoidally frequency-modulated signal with low modulation index:

application to laser measurements in acoustics," Signal Processing, vol. 84,

no. 7, pp. 1147 - 1165, Jul. 2004.

[7] M. Glickman and P. O'Shea, "Damping estimation of electric disturbances in

distributed power systems," 7th IASTED Int. Conf. Sig. and Image Proc.,

Aug. 2005.

[8] M. Glickman, P. O'Shea, and G. Ledwich, "Damping estimation in highly

interconnected power systems," IEEE Region 10 - TENCON '05, Nov. 2005.

[9] M. Glickman, P. O'Shea, and G. Ledwich, "Estimation of modal damping in

power networks," IEEE Trans. Power Syst., vol. 22, no. 3, pp. 1340 - 1350,

Aug. 2007.

[10] P. O'Shea, "Detection and estimation methods for non-stationary signals,"

PhD Dissertation, University of Queensland, Brisbane, Australia, 1991.

[11] P. O'Shea, "Fast parameter estimation algorithms for linear FM signals,"

Proc. IEEE Int. Conf. Acoust., Speech, and Sig. Proc. (ICASSP), vol. 4, Apr.

1994.

[12] P. O'Shea, "An iterative algorithm for estimating the parameters of

polynomial phase signals," Proc. IEEE 4th Int. Symposium Sig. Proc. and Its

Appl. (ISSPA), vol. 2, pp. 730 - 731, Aug. 1996.

[13] P. O'Shea, "The use of sliding spectral windows for parameter estimation of

decaying sinusoidal signals," Proc. IEEE Region 10 - TENCON '97 Annu.

Conf. Speech and Image Techn. for Computing and Telecomms., vol 2, pp.

827 - 830, Dec. 1997.


IEEE Int. Conf. Acoust., Speech, and Signal Processing (ICASSP), vol. 6, pp.

3570 - 3573, Jun. 2000.

[15] P. O'Shea, "The use of sliding spectral windows for parameter estimation in

power system disturbance monitoring," IEEE Trans. Power Syst., vol. 15, no.

4, pp. 1261 - 1267, Nov. 2000.


48

[16] P. O'Shea, "A new technique for instantaneous frequency rate estimation,"

IEEE Sig. Proc. Letters, vol. 9, no. 8, pp. 251 - 252, Aug. 2002.

[17] P. O'Shea, "A high-resolution spectral analysis algorithm for power-system

disturbance monitoring," IEEE Trans. Power Syst., vol. 17, no. 3, pp. 676 -

680, Aug. 2002.

[18] P. O'Shea, "A fast algorithm for estimating the parameters of a quadratic FM

signal," IEEE Trans. Sig. Proc., vol. 52, no. 2, pp. 385 - 393, Feb. 2004.

[19] P. O'Shea, M. J. Arnold, and B. Boashash, "Some techniques for estimating

the instantaneous frequency of rapidly time-varying signal," IEEE Sig. Proc.,

Theories, Implementation and Appl. (ISSPA) Conf., 1990.

[20] P. O'Shea, M. L. Farquharson, and G. Ledwich, "Estimation of time-varying

mains frequencies," Proc. AUPEC Conf., Sep. - Oct. 2003.





5, pp. 2968 - 2972, Jun. 1998.



1, pp. 226 - 231, Feb. 1999.

[24] C. L. Zhang and G. F. Ledwich, "A new approach to identify modes of the

power system based on T-matrix," Proc. 6th IEEE Int. Conf. Advances in

Power Syst. Control, Operation and Management (APSCOM), vol. 2, pp. 496

- 501, Nov. 2003.

[25] R. Kumaresan and D. Tufts, "Estimating the parameters of exponentially

damped sinusoids and pole-zero modeling in noise," IEEE Trans. Acoust.,

Speech, and Sig. Proc., vol. 30, no. 6, pp. 833 - 840, Dec. 1982.

[26] D. Rife and R. Boorstyn, "Single tone parameter estimation from discrete-

time observations," IEEE Trans. Inform. Theory, vol. 20, no. 5, pp. 591 - 598,

Sep. 1974.


49

[27] T. I. Salsbury and A. Singhal, "A new approach for ARMA pole estimation

using higher-order crossings," Proc. American Control Conf., vol. 7, pp. 4458

- 4463, Jun. 2005.

[28] G. E. P. Box, G. M. Jenkins, and G. C. Reinsel, Time series analysis:

forecasting and control, 3rd ed. Englewood Cliffs, N.J.: Prentice Hall, 1994.

[29] J. D. Hamilton, Time series analysis. Princeton, N.J.: Princeton University

Press, 1994.

[30] L. Ljung, System identification: theory for the user. Englewood Cliffs, NJ:

Prentice-Hall, 1987.

[31] B. Friedlander and K. Sharman, "Performance evaluation of the modified

Yule-Walker estimator," IEEE Trans. Acoust., Speech, and Signal Proc., vol.

33, no. 3, pp. 719 - 725, Jun. 1985.

[32] Y.-C. Liang, X.-D. Zhang, and Y.-D. Li, "A hybrid approach to time series

analysis and spectral estimation," Proc. American Control Conf., vol. 1, pp.

124 - 128, Jun. 1995.

[33] R. Moses, P. Stoica, B. Friedlander, and T. Soderstrom, "An efficient linear

method for ARMA spectral estimation," Proc. IEEE Int. Conf. Acoust.,

Speech, and Sig. Proc. (ICASSP), vol. 12, pp. 2077 - 2080, Apr. 1987.

[34] J.-H. Hong and J.-K. Park, "A time-domain approach to transmission network

equivalents via Prony analysis for electromagnetic transients analysis," IEEE


[35] M. A. Johnson, I. P. Zarafonitis, and M. Calligaris, "Prony analysis and

power system stability-some recent theoretical and applications research,"

Proc. IEEE Power Eng. Soc. Summer Meet., vol. 3, pp. 1918 - 1923, Jul.

2000.

[36] M. Amono, M. Watanabe, and M. Banjo, "Self-testing and self-tuning of

power system stabilizers using Prony analysis," Proc. IEEE Power Eng. Soc.

Wint. Meet., vol. 1, pp. 655 - 660, Jan. - Feb. 1999.

[37] M. Meunier and F. Brouaye, "Fourier transform, wavelets, Prony analysis:

tools for harmonics and quality of power," Proc. IEEE 8th Int. Conf.

Harmonics And Quality of Power, vol. 1, pp. 71 - 76, Oct. 1998.


50

[38] P. S. Dolan, J. R. Smith, and W. A. Mittelstadt, "Prony analysis and modeling

of a TCSC under modulation control," Proc. 4th IEEE Conf. Control Appl.,

pp. 239 - 245, Sep. 1995.

[39] P. K. Gale and J. W. Pierre, "Prony analysis based parameter estimation of an

NMR signal of blood plasma for cancer detection," Proc. IEEE Int. Conf.

Acoust., Speech, and Sig. Proc. (ICASSP), vol. 2, pp. 1185 - 1188, May 1995.

[40] J. C. Mosher and P. S. Lewis, "Taylor series expansion and modified

extended Prony analysis for localization," Proc. IEEE 28th Asilomar Conf.

Sig., Syst. and Computers, vol. 1, pp. 667 - 670, Oct. - Nov. 1994.

[41] C. E. Grund, J. J. Paserba, J. F. Hauer, and S. L. Nilsson, "Comparison of

Prony and eigenanalysis for power system control design," IEEE Trans.

Power Syst., vol. 8, no. 3, pp. 964 - 971, Aug. 1993.

[42] J. F. Hauer, C. J. Demeure, and L. L. Scharf, "Initial results in Prony analysis

of power system response signals," IEEE Trans. Power Syst., vol. 5, no. 1,

pp. 80 - 89, Feb. 1990.

[43] J. F. Hauer, "Application of Prony analysis to the determination of modal

content and equivalent models for measured power system response," IEEE

Trans. Power Syst., vol. 6, no. 3, pp. 1062 - 1068, Aug. 1991.

[44] D. A. Pierre, D. J. Trudnowski, and J. F. Hauer, "Identifying linear reduced-

order models for systems with arbitrary initial conditions using Prony signal

analysis," IEEE Trans. Automatic Control, vol. 37, no. 6, pp. 831 - 835, Jun.

1992.

[45] A. A. Beex and P. Shan, "A time-varying Prony method for instantaneous

frequency estimation at low SNR," Proc. IEEE Int. Symposium Circuits and

Syst. (ISCAS), vol. 3, pp. 5 - 8, May. - Jun. 1999.

[46] D. Tufts and R. Kumaresan, "Singular value decomposition and improved

frequency estimation using linear prediction," IEEE Trans. Acoust., Speech,

and Sig. Proc., vol. 30, no. 4, pp. 671 - 675, Aug. 1982.

[47] M. D. Rahman and K.-B. Yu, "Total least squares approach for frequency

estimation using linear prediction," IEEE Trans. Acoust., Speech, and Sig.

Proc., vol. 35, no. 10, pp. 1440 - 1454, Oct. 1987.


51



Jan. 2000.

[49] J. J. Sanchez-Gasca, "Computation of turbine-generator subsynchronous

torsional modes from measured data using the eigensystem realization

algorithm," Proc. IEEE Power Eng. Soc. Wint. Meet., 2001.

[50] K. E. Bollinger and W. E. Norum, "Time series identification of interarea and

local generator resonant modes," IEEE Trans. Power Syst., vol. 10, no. 1, pp.

273 - 279, Feb. 1995.

[51] D. Ruiz-Vega, M. Pavella, and A. R. Messina, "On-line assessment and

control of poorly damped transient oscillations," Proc. IEEE Power Eng. Soc.

Gen. Meet., vol. 4, pp. 2084 - 2089, Jul. 2003.

[52] M. J. Gibbard, N. Martins, J. J. Sanchez-Gasca, N. Uchida, V. Vittal, and L.

Wang, "Recent applications of linear analysis techniques," IEEE Trans.

Power Syst., vol. 16, no. 1, pp. 154 - 162, Feb. 2001.

[53] D. Ruiz-Vega, A. R. Messina, and M. Pavella, "An Approach to On-line

Assessment of Power System Damping," 14th Power Syst. Computation

Conf. (PSCC), Jun. 2002.

[54] S. M. Kay, Modern spectral estimation: theory and application. Englewood

Cliffs, N.J.: Prentice-Hall, 1988.

Chapter 3: Multiple Orthogonal

Window Estimation Method

3.1 Introduction Distributed power system generators are interconnected to share power and to meet

the demand. During normal power system operation oscillating modal components

arise due to line tripping, changes in the load and various faults. It is important to

monitor these oscillations to ensure that there are no exponentially growing modes

(corresponding to “negative damping”). These negative damping modes can cause

the system to fail. Regular monitoring of modes (with damping estimation strategies)

will allow detection of dangerously low (or negative) damping factors and will give

enough time for introduction of appropriate control measures.

The oscillating modes are assumed to have the form:

( ) ( ) ( ) ( ) ( )nAennznz nnjsr εε αφω +=+= −+0 (3.1)

where =A Amplitude, =α damping, =0ω modal frequency (rad/s), =φ initial phase

(rad), ( ) =nε additive complex Gaussian noise. Recall that modal signals in practice

are real, but the complex form of the mode can be obtained by forming the analytic

signal.

There are a number of methods for estimating the damping of power system

observation records [1-10]. The sliding window method is one of those techniques. It

estimates the damping via Fourier transformation (FT) of two consecutive time

windows [1-7]. The method is based on the fact that the logarithmic ratio of Fourier

amplitudes is proportional to the rate of modal decay. Basic sliding window methods

compute the damping by calculating the logarithmic ratio between the FTs of two


53

sampled sequences. Normally the two sequences are multiplied by window functions

sequence before the FFT is computed, mainly so as to reduce the interference

between spectral sidelobes [7]. Different choices for the applied windows will result

in different estimation accuracy. The number of samples between windows also

affects the accuracy of damping estimates, with the formula for the optimum number

of samples is derived in [7].

The second type of damping estimation approach is to use parameter estimation

techniques. Prony method is a well known parametric estimation method used for

power system disturbance detection. The algorithm models the electric disturbance

signals as an output of a linear time invariant (LTI) system connected to a white

Gaussian noise (WGN) generator. Modal parameters are estimated by solving a set of

LTI equations [8, 9], as outlined in [10, 11]. The main advantages of the algorithm

are high resolution that allows estimation of closely spaced modes, a capability

which standard sliding window methods do not have. However, Prony techniques do

not work well at low SNRs. Various improvements to Prony methods have been

suggested in [9, 10, 12] to improve the numerical conditioning and the accuracy [9].

The first modification suggested Kumaresan and Tufts in [9] was to solve the linear

prediction matrix equation with Singular Value Decomposition (SVD) instead of

traditional matrix inversion techniques. The second proposed modification was to use

a high order of linear prediction model, L . It should be noted, however, that there is

a trade-off – increasing the order does improve accuracy but it also increases

computation time. Simulations show that the Kumaresan-Tufts (KT) enhanced Prony

analysis yields lower MSEs than Prony methods. See, for example, Figure 3.1.

Chapter 3: Covariance Least Squares Averaging

54

6 8 10 12 14 16 18 20 22 24 26 -90 -80 -70 -60 -50 -40 -30 -20 -10

0M

ean

squa

re e

rror

in d

ampi

ng (d

B)

SNR (dB) Figure 3.1 Comparison of KT (full) and basic Prony (dashed) damping

estimate MSEs for a single mode [5].

Damping estimation for closely spaced modes presents a significant challenge.

Standard Fourier methods are not effective because closely spaced modes appear in

the spectrum to be just “one mode”. Parametric methods such as Prony method are

more effective but still require appropriate orders to be selected. Prony’s method

would only estimate the average frequencies and the overall damping of the group if

the number of modes assumed is lower than the number of modes present. The first

part of this chapter deals with the case where there are no closely spaced modes and

so Fourier techniques alone are used. A sliding window algorithm is presented that

uses the multiple orthogonal sliding windows of Slepian [13] to compute multiple

damping estimates. These multiple damping estimates are combined with least

squares averaging techniques. (Note that the multiple damping estimates are

statistically independent because the windows are orthogonal). The multiple damping

estimates obtained from the K windows can also be used for cross validation of

results. The latter part of the chapter deals with the case where closely spaced modes

are present and a combination of the multiple orthogonal window method and

parametric techniques are used. The work in this chapter has been reported in [5].


55

3.2 Multiple Orthogonal Window Damping Estimation

Method 3.2.1 Background: Basic Sliding Window Method The first step in the basic sliding window methods is frequency estimation:

( ) ⎥⎦

⎤⎢⎣

⎡= ∑

−

=

−1

00 maxargˆ

N

n

njr enz ω

ωω (3.2)

It is important to note that the above frequency estimation process assumes the

frequency does not vary appreciably over the time of the measurement. Modal

frequency drifts do occur with time due to such things as load changes [10]. Methods

for estimating modal parameters with change in frequencies are presented in [14]. It

is best, however, to wait till the system stabilizes before applying the algorithms

presented in this paper. Methods for detecting when changes are occurring in

important power system parameters are presented in [15-21].

The second step in the basic sliding window method is to sample the input signal

with two windows, applied at different time positions, to obtain:

( ) ( ) ( ) ( ) ( )nwnnznwnznz srr ε+== )(11

10 −≤≤ wNn (3.3)

and

( ) ( ) ( ) ( ) ( )nwNnnznwNnznz gsgrr ++=+= ε)(22 10 −≤≤ wNn (3.4)

where Number of samples between windows, =gN =wN Number of samples in a

window. In [1, 2] the window was restricted to being rectangular while in [7] smooth

windows were used to reduce the interference between spectral magnitude peaks.

Rectangular windows (used in [1]) start and end rapidly whereas smooth windows

(applied in [7]) taper slowly on and off. The advantage of using smooth windows is


56

low Fourier magnitude side lobes and thus low spectral influence on other modes.

The differences between rectangular and smooth windows are illustrated in Figure

3.2. Spectral side lobes, which are apparent in Figure 3.2 (b), can be mistaken for

other oscillating modal components, thus causing problems for the analysis of

multiple modes. This interference due to side-lobes is a significant problem even for

single real modes because the positive and negative frequency components tend to

interfere [7].

0 200 4000

0.5

1

Time (seconds)

w(t)

0 0.02 0.040

0.5

1

Frequency (Hz)

|W(f

)|

(a) (b)

0 200 4000

0.5

1

Time (seconds)

w(t)

0 0.02 0.040

0.5

1

Frequency (Hz)

|W(f

)|

(c) (d)

Figure 3.2 (a) Rectangular window. (b) FT magnitude of a rectangular window. (c) Smooth (Kaiser)

window. (d) FT magnitude of a smooth (Kaiser) window [5].

The third step in the basic sliding window method is to estimate the damping


57

according to:

( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

++

= −202

1010

Relog1ˆqeZ

qZN gNj

s

s

gωω

ωα (3.5)

where )( 01 ωsZ and )( 02 ωsZ are the windowed signal FTs of and ( )nzs1 ( )nzs2

respectively; and are the perturbations to the spectral samples due to the noise

on the observation; Re(.) denotes the real part. It is assumed that noise spectral

magnitudes are small compared to oscillation spectral magnitudes and that damping

estimate bias is very close to zero. Mean square error (MSE) or variance of damping

factor estimate is equal to:

1q 2q

( ) [ ]2|ˆ|)ˆ(ˆvar αααα −== EMSE (3.6)

where is the expected value. The variance for [ ].E α is derived in the appendix.

The complex amplitude estimate can be computed with the formula below:

( )

( )∑−

=

−

= 1

0

0ˆ 1ˆwN

n

n

sj

enw

ZeA

α

φ ω (3.7)

3.2.2 The Sliding Multiple Window Method One of the ways of reducing the variance is by optimizing the window length [7].

Further reductions in variance can be achieved with the use of multiple orthogonal

sliding windows [3, 5]. The “orthogonal window” concept will be explained in the

following paragraphs.

The orthogonal windows of Slepian are a group of windows which have zero

correlation with one another in both the time and frequency domains. Windows are


58

generated with an eigen decomposition process as presented in [22]. This process can

be realized in MATLAB with the “DPSS” instruction in the Signal Processing

Toolbox. Five orthogonal windows generated with 4=BNw (where ( )π22B is the

window bandwidth) are shown in Figure 3.3. The plots show that the lower order

windows de-emphasize data at the start and end of the record. Combining estimates

from all windows will allow data from all parts of the record to be efficiently used.

0 200 400-1

-0.5

0

0.5

1

Time (seconds)

w1(t)

0 0.02 0.040

0.5

1

Frequency (Hz)

|W1(f

)|

(a)

0 200 400-1

-0.5

0

0.5

1

Time (seconds)

w2(t)

0 0.02 0.04

0.2

0.4

0.6

0.8

1

Frequency (Hz)

|W2(f

)|

(b)

0 200 400-1

-0.5

0

0.5

1

Time (seconds)

w3(t)

0 0.02 0.040

0.5

1

Frequency (Hz)

|W3(f

)|

(c)


59

0 200 400-1

-0.5

0

0.5

1

Time (seconds)

w4(t)

0 0.02 0.04

0.2

0.4

0.6

0.8

1

Frequency (Hz)

|W4(f

)|

(d)

0 200 400-1

-0.5

0

0.5

1

Time (seconds)

w5(t)

0 0.02 0.040

0.5

1

Frequency (Hz)

|W5(f

)|

(e)

Figure 3.3 Five orthogonal windows and their FTs ( ) [5]. 4=BN w

The acquired observation sequence can be sampled with K pairs of orthogonal

windows to yield K damping factor estimates. Those K pairs of windows are

defined below:

( ) ( ) ( )nwnznz ksks =1

(for 10 −≤≤ wNn ) (3.8)

and

( ) ( ) ( )gksks Nnwnznz −=2

(for 1−+≤≤ wgg NNnN ) (3.9)

where is the orthogonal window. Because of the orthogonality of the

windows damping estimates are un-correlated if the noise spectrum is flat over the

)(nwkthk


60

(relatively narrow bandwidth of Bπ4 . A typical B value is . The uncorrelated

nature of the estimates allows the use of least squares averaging to yield an ‘overall’

damping factor estimate with reduced variance [3, 5]. The ‘overall’ damping factor

estimate is determined from optimal estimation theory as [23]:

wN/4

( ) rCXXCX TT 111ˆ −−−=α (3.10)

where , and is the covariance matrix for [ TX 111 K= ] C r .

C is equals to:

( )( )[ ] ( )( )[ ]

( )( )[ ] ( )( )[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−

−−−−=

KKK

K

EE

EEC

αααααααα

αααααααα

ˆˆˆˆ

ˆˆˆˆ

1

111

L

MOM

L

(3.11)

Because of the orthogonality of the windows all estimates in r vector are

uncorrelated. All elements in the covariance matrix except those on the diagonal will

be equal to zero:

( )

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡≈

K

Cα

α

ˆvar0

0ˆvar 1

L

MOM

L

(3.12)

where )ˆvar( kα is the theoretical MSE equation derived in the appendix. The variance

elements are also obtained from the theoretically derived MSE equation in the

appendix. The variance of the least squares average damping estimate is known from

optimal estimation theory to be:

( ) ( ) 11ˆvar −−= XCX Tα (3.13)


61

3.2.3 Summary of the Sliding Multiple Window Method Step 1: Determine the frequency of the oscillating mode from equation (3.2):

Step 2: Sample K pairs of window with orthogonal windows defined in (3.8) and

(3.9).

Step 3: Form K damping factor estimates:

( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛= − gNjk

r

kr

gk eZ

ZN 0

02

01

ˆˆ

Relog1ˆωω

ωα

(for Kk ,,2,1 K= ) (3.14)

where and are the FTs of ( 01 ωkrZ ) )( 02 ω

krZ ( )nz k

r1 and ( )nz kr 2 respectively.

Step 4: Calculate the least squares average damping estimate via equations (3.10)

and (3.11).

The method in (3.10-3.12) requires knowledge of α which is not available a priori.

An initial damping estimate can be computed, however, by using just the first

orthogonal window. An alternative is to not use (3.10-3.12) to compute the overall

damping estimate but instead divide each of the orthogonal window damping

estimates, kα by the mean value of the corresponding window and then average the

results obtained. Complex amplitude estimates are found from (3.7).


62

3.3 Simulations to Compare Multiple Sliding Window

Method with Basic Sliding Window Method 3.3.1 Additive White Noise A sequence of the form in (3.1) was generated with additive white Gaussian noise

(AWGN) for various noise powers. The following simulation parameters were used:

number of samples, ; amplitude, 512=N 1=A ; phase, 0=φ rad; damping,

005.0=α ; frequency, 982.00 =ω rad/s; number of windows in the sliding multiple

window method, 5=K ; window bandwidth, 4=BNw Hz; number of samples

between windows and the number of samples in both windows,

samples. A basic sliding window method was simulated with only the first order

orthogonal window [22], and a multiple sliding window method with 5 orthogonal

windows was also simulated. For Figures 3.4 and 3.5 the multiple sliding window

method damping estimate MSEs are depicted with circles, the basic sliding window

method damping estimate MSEs are depicted with dots and the theoretical minimum

damping estimate MSEs (Cramer-Rao bounds) are shown as a dashed line. Results

show that multiple sliding window method yields ~2.5 dB lower MSE damping

estimates than the basic sliding window method.

64== wg NN

3.3.2 Additive Coloured Noise Simulations were also performed for additive coloured Gaussian noise simulations.

Coloured noise occupying the same band as the signal was generated by passing

white noise through a narrowband filter (bandwidth equal to 1/16 of the total band).

The noise was therefore flat for 1/16 of the total available band and zero elsewhere.

Simulations showed that there was virtually no observable difference between the

white and colored noise scenarios. This is not surprising because the smooth

windows have low side lobes and suppress the noise outside the (narrow) spectral

region of interest. Thus the influence of noise and other modes is suppressed in the

Fourier domain when the smooth windows are used. This indicates the new multiple

sliding window method is useful for both white and coloured noise.


63

6 8 10 12 14 16 18 20 22 24 26-100 -95 -90 -85 -80 -75 -70 -65 -60 -55 -50

SNR (dB)

Mea

n sq

uare

err

or in

dam

ping

(dB

)

Figure 3.4 Comparison of Basic Sliding Window Method (dots), Multiple Sliding Window Method

(circles) and Cramer-Rao bound (dotted line) [5].

3.4 High Resolution Multiple Window Method The algorithm in Section 3.2 is an improvement on the basic sliding window method

but it cannot estimate the damping of multiple closely spaced modes because they are

appear as one mode in the Fourier spectrum. The second problem with the method in

Section 3.2 is that the sliding window method is evaluated at two different time

positions, limiting the accuracy of estimates. The extended method, known as the

high resolution sliding window method was devised to solve those problems. The

high resolution sliding window method was first introduced in [24] and is able to:

(i) process multiple closely spaced modes,

(ii) achieve accurate damping estimates,

(iii) estimate damping factors in a computationally efficient manner.


64

To achieve goal i) the input sequence is sampled with more than two windows. This

allows parameter estimation of multiple modes.

In order to achieve goal ii) the filtering must be done as shrewdly as possible to

prevent distortion and loss of information at the onset of the mode (where SNR is the

highest). Thus filters with good frequency selectivity and short impulse response

should be used. The orthogonal windows of Slepian were designed for this purpose.

Those functions where derived for optimal bandwidth concentration for a given

impulse response duration [13]. The use of sliding Slepian functions is thus an

effective filtering technique for short duration sequences.

Goal iii) is achieved with the use of SPECTROGRAM instruction in MATLAB’s

Signal Processing Toolbox. This instruction computes the sliding Fourier transform

of acquired data. Thus the output consists of sliding channels of filtered data. This

operation is very efficient because a large number of filtering operations are done

simultaneously, thus reducing the time it takes to compute parameters of multiple

modes. The second strategy is the use of “demodulation to baseband” so that lower

sampling rates can be used. SPECTROGRAM outputs are narrowband (because of

the filtering inherent in the sliding FFTs) and so one can reduce the sampling by a

factor ; then reduction in computation will fall by . The least square average

method computation time will be

G 3G

K times that of the basic sliding window method

because spectrogram is computed for each window.

Steps of the high resolution sliding window method can be presented in the form of

mathematical equations:

Step 1: Evaluate K sliding window Fourier transforms, ( )ω,nFk :

( ) ( ) ( ) (∑−

=

−+=1

0exp,

wN

mrk mjmwnmznF ωω ) (3.15)


65

where =ω Frequency of FFT sample; gvNn = ; Vv ,,2,1,0 K= , where 1+> MV ,

. Kk ,,1K= ( )ω,nFk is created with MATLAB SPECTROGRAM command.

Step 2: Determine the initial frequency estimate for the modal cluster of interest. The

initial frequency is the frequency corresponding to the maximum Fourier transform

magnitude in the cluster of interest. Let M be the number of frequencies in the

modal cluster.

Step 3: “Demodulate” the sliding window time series from Step 1 by downshifting

the frequency region to baseband (i.e. the 0 Hz region):

( ) ( ) ( )njnFnF iikdk 00 ˆexpˆ, ωω −= , Kk ,,1,0 K= (3.16)

Step 4: The resulting sequence is fed in to the KT method for damping and

frequency estimation of the M downshifted modes. A weighted average of the K

damping factor estimates is then performed to obtain an overall damping estimate:

∑

∑

=

== K

kk

K

kkm

m

u

uk

1

1ˆα

α , Mm ,,1,0 K= (3.17)

where is the average of orthogonal windows; ku thkkmα is the damping estimate of

mode and orthogonal window. thm thk mα is the overall damping estimate for

mode, determined via the weighted average procedure in (3.17), rather than with

covariance based procedure (3.10-3.12) because reliable covariance information is

difficult to determine for multiple modes.

thm


66

In the case of data from multiple sites, linear prediction equations can be created for

each channel of multiple site data and combined in a single matrix equation which is

then solved via SVD to obtain the overall damping estimate [10, 12].

3.5 Simulations In what follows the high resolution multiple window method is tested for its

performance in a number of different scenarios. SNR and damping factor were varied

to investigate their effect on MSE in these various scenarios. In all cases the

orthogonal window bandwidth was set to 0.1811 rad/sample. 200 simulation trials

where done for each set of parameters.

3.5.1 Scenario I. A Single Mode in White Noise. Damping Factor is

Held Constant and SNR is Varied The following simulation parameters where used: number of modes, 1=M ; number

of samples in a window and step time, 56== wg NN samples; amplitude, 1=A ;

damping factor, 005.0=α ; frequency equal to 0.982 rad/s; phase equal to 0 rad;

; order of linear prediction in high resolution sliding multiple window method,

; order of linear prediction model in KT method,

9=V

5=HRMWL 350=KTL ; KT method

data length, and noise was additive white Gaussian. 500=KTN

Simulation results are plotted in Figure 3.5(a) and (c), with damping MSE plotted

versus SNR. The high resolution multiple sliding window method MSE is seen to be

a little better than the high resolution basic sliding window method. Both of these

methods outperform the KT method at low SNR. A typical realization of the noisy

signal is shown in Figure 3.5(b).


67

6 8 10 12 14 16 18 20 22 24 26

-90 -80 -70 -60 -50 -40 -30 -20

SNR (dB)

Mea

n sq

uare

err

or in

dam

ping

(dB

)

Figure 3.5(a) Damping MSE vs SNR for high resolution basic sliding window (dots), high resolution

multiple sliding window (circles) and KT methods (full line), . The dotted line represents

the Cramer-Rao (CR) lower variance bound [5].

500=KTL

0 100 200 300 400 500 600-1

-0.5

0

0.5

1

Rea

l par

t of o

bser

vatio

n

Sample number Figure 3.5(b) Input sequence, noisy time domain signal [5].


68

One can reduce the SNR threshold for the KT method by increasing the KT linear

prediction model, order, as shown in Figure 3.5(c). This, however, has the effect

of increasing the MSE.

KTL

6 8 10 12 14 16 18 20 22 24 26 -100 -90 -80 -70 -60 -50 -40 -30

Mea

n sq

uare

err

or in

dam

ping

(dB

)

SNR (dB)Figure 3.5(c) Damping MSE vs SNR for high resolution basic sliding window (dots), high resolution



300=KTL

3.5.2 Scenario II. A Single Mode in White Noise. SNR is Held

Constant and Damping Factor is Varied The following parameters were used for simulations: 10=SNR dB; number of

samples, ; number of modes, 500=N 1=M ; amplitude, 1=A ; damping factor was

varied from 0.005 to 0.025; frequency, 982.00 =ω rad/s and phase, 0=φ rad;

; window length and step time, 9=V == gw NN round(56*scale factor), where scale

factor ( )( ) 500//0821.0log 2σ= , KT method linear prediction model order =

round(350*scale factor), order of linear prediction in high resolution multiple sliding


69

window method = round(K*scale factor) and noise was additive white Gaussian.

Expression round(.) represents rounding to the nearest integer.

Simulation results in Figure 3.5(d) show that the KT method is more vulnerable to

failure. This is not entirely surprising – parametric methods have a reputation for

being less robust, particularly at low SNRs.

0.005 0.01 0.015 0.02 0.025-80

-70

-60

-50

-40

-30

-20

-10

Damping factor

Mea

n sq

uare

err

or in

dam

ping

(dB

)

Figure 3.5(d) Damping MSE vs damping for high resolution basic sliding window (dots), high

resolution multiple sliding window (circles) and KT methods (full line), [5]. 500=KTL

3.5.3 Scenario III. Two Closely Spaced Modes in White Noise. SNR

is Varied. Simulation parameters were set to: number if modes, 2=M ; window length and

step time, ; amplitudes equal to 1; 56== gw NN 9=V ; high resolution sliding

window method linear prediction model order, 5=HRMWL ; KT method linear


70

les in KT

he noiseless sign n in Figure 3.5(e). Damping factor estim

prediction model order, 180=KTL ; number of samp method, 300=KTN ;

damping factors equal to 0.005 and 0.001 respectively; frequencies equal to 0.982

and 1.104 rad/s respectively, phase equal to 0 rad and noise was additive white

Gaussian.

T al is show ates of first

mode (damping equal to 0.005) are plotted in Figure 3.5(f). The high resolution

sliding multiple window method has the lowest damping MSEs for this set of

simulations.

Figure 3.5(e) Input sequence, noiseless time domain signal [5].

0 100 200 300 400 500-2

-1.5 -1

-0.5 0

0.5 1

1.5 2

Sample number

Am

plitu

de

600


71

6 8 10 12 14 16 18 20 22 24 26

-80 -75 -70 -65 -60 -55 -50 -45 -40 -35

SNR (dB)

Mea

n sq

uare

err

or in

dam

ping

(dB

)

-85 -90

Figure 3.5(f) Damping MSE vs SNR for high resolution basic sliding window (dots), high resolution



300=KTL

3.5.4 Scenario VI. Two Heavily Damped Modes in White Noise. SNR

is Varied The following simulation parameters were used: number of modes, 2=M , number

of samples in a window, 32=wN samples; step time, 4=gN samples; ; high

resolution sliding window method linear prediction model order, ; KT

method linear prediction model order,

9=V

5=HRMWL

30=KTL ; number of samples in KT method,

; amplitudes equal to 1; damping factors equal to 0.05 and 0.075 60=KTN


72

respectively, frequencies equal to 0.982 and 2.356 rad/s respectively, phases equal to

0 rad.

The noiseless signal is shown in Figure 3.5(g). Damping factor estimates of the first

mode (damping equal to 0.05) are plotted in Figure 3.5(h). The KT method is seen to

have the lowest damping MSE because heavily damped modes (that decay within a

short period of time) are not well filtered with sliding window methods.

0 20 40 60 80 100 120 140 160 180 200 -0.5

0

0.5

1

1.5

2

Sample number

Am

plitu

de

Figure 3.5(g) Input sequence, noiseless time domain signal [5].


73

35 40 45 50 55-100

-85

-80

-75

-70

-65

-60

SNR (dB)

Mea

n sq

uare

err

or in

dam

ping

(dB

)

-95

-90

Figure 3.5(h) Damping MSE vs SNR for high resolution basic sliding window (dots), high resolution



300=KTL

Simulation results in this section show that the sliding multiple window method is

not the best method for modal parameter estimation in all scenarios. For this reason it

is recommended that the new method be used as one of a number of possible tools.


74

3.6 Application to a Simulated Power System Example and

Real Power System Data 3.6.1 Simulated Power System Example A simulated power system model consisting of four generators interconnected with

impedances between nodes i and j denoted by is shown in Figure 3.6. ijX

X12=0.015 X23=0.018

J1=10 J2=16 J3=14 J4=4.5

X34=0.02

P1 P2 P4P3 Figure 3.6 Model of simulated power system [5].

Machine model parameters are: inertia, ; machine voltages, and ; input

power, ; load power, and angle of internal voltages (classical machine model

is assumed),

iJ iV jV

imP iP

iδ . The power system model response can be represented in the

following equation:

( )∑

−−−=

j ij

jijiimii X

VVPPJ

i

δδδ

sin&& (3.18)

By assuming that:

• input powers are equal to the mean of local powers,


75

• load power variations are modeled as a random process obtained by integrating

white noise,

• pu, 1== ji VV

• differences between machine angles are small such that ( ) δδ ≈sin .

the power system model response will equal:

( )∑

−−Δ−≈

j ij

jijiiii X

VVPJ

δδδ&& (3.19)

This simple model is sufficient to approximate the Australian power system

response. Angle variations were such that there was a standard deviation of about 1

degree from the quasi-stationary operating point. Those variation levels are similar to

the ones observed in Australian power system. The measurements at Adelaide,

Brisbane, Melbourne and Sydney do not show local angle oscillations. Hence they

behave as single machines in each state. The impedances and inertias are roughly the

same in each of the four Australian states. 3 hours of 10 samples per second data was

generated in MATLAB (the code is provided in the appendix) based on the simulated

system described above. The autocorrelation function obtained from one of the

simulated generators is shown in Figure 3.7 and simulations results are shown in

Table 3.1 and Table 3.2.


76

0 2 4 6 8 10 12-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3x 10-3

Time (seconds)

Rea

l com

pone

nt o

f obs

erve

d si

gnal

Re[

z r(t)

]

Figure 3.7 Autocorrelation of data generated by the simulated power system [5].

0 0.2 0.4 0.6 0.8 1 1.2

0.005

0.01

0.015

0.02

0.025

0.03

Frequency (Hz)

Four

ier t

rans

form

mag

nitu

de o

f obs

erve

d si

gnal

, |Z

r(f)|

Figure 3.8 Fourier transform magnitude of observed signal [5].


77

As illustrated by the data in Table 3.1 and Table 3.2, the new high resolution sliding

multiple window method is seen to perform comparatively well in this example.

Damping estimates

Mode 1 Mode 2 Mode 3

True values 0.4768 0.1868 0.5070

Basic sliding window method 0.4672 0.1871 0.5060

High resolution sliding multiple window method 0.4768 0.1871 0.5070 Table 3.1 Damping estimates [5].

Damping MSE (dB)

Mode 1 Mode 2 Mode 3

Basic sliding window method -61.4099 -68.3039 -66.3011

High resolution sliding multiple window method -89.4409 -69.4026 -93.4376Table 3.2 Damping estimates MSE [5].

3.6.2 Real Power System Example Real power data was acquired from Blackwall substation in near Brisbane in

Queensland, Australia. The oscillation was initiated by a 300 MW braking resistor at

Gladstone power station 800 km away. The resistor was switched for 0.2 seconds.

Measurements of the inter-area mode between Queensland and Southern States were

done at a distance to reduce the affect of local disturbances and oscillations. The

sampling rate was 5 samples per second. The autocorrelation function of the acquired

data and it spectrum are shown in Figure 3.9 and Figure 3.10 respectively. The larger

modal component (0.3414 Hz) is the inter-area mode.


78

0 5 10 15 20 25

-10

-5

0

5

10

15

20

Time (seconds)

Rea

l com

pone

nt o

f obs

erve

d si

gnal

Re[

z r(t)

]

Figure 3.9 Input sequence – time domain autocorrelation of acquired real data.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

20

40

60

80

100

120

140

160

180

200

220

Frequency (Hz)

Four

ier t

rans

form

mag

nitu

de o

f obs

erve

d si

gnal

, |Z

r(f)|

Figure 3.10 Fourier transform magnitude of input sequence.


79

The non-high resolution sliding window algorithm was implemented. Frequency,

damping, amplitude and phase estimates for the two modes were used to generate an

estimate of the input sequence. The residual between the estimated signal and the

original signal was calculated. The lowest residual power was obtained for sliding

multiple window method.

Method Residual (dB)

Basic sliding window method -10.5630 Sliding multiple window method -11.4828

Table 3.3 Residual between acquired and estimated signals [5].

3.7 Conclusion The new Fourier based multiple sliding window methods have been presented for

estimating the modal damping. This method uses multiple orthogonal sliding

windows and least-squares error minimization techniques. Simulations show that the

algorithm can estimate the damping with lower MSE than the basic sliding window

or Prony methods for various scenarios. The new methods do not outperform the KT

method in all situations and should therefore be used as one of a number of possible

tools. The new method also worked well when applied to real data.

3.8 References [1] K. K.-P. Poon and K.-C. Lee, "Analysis of transient stability swings in large





vol. 5, no. 1, pp. 148 - 153, Feb. 1990.



Aug. 2005.


80





Aug. 2007.



Conf. Speech and Image Techn. for Computing and Telecomms., Dec. 1997.



4, pp. 1261 - 1267, Nov. 2000.








1, pp. 226 - 231, Feb. 1999.

[11] B. P. Administration, "Internet site, http://www.bpa.gov.au/corporate," 2006.



5, pp. 2968 - 2972, Jun. 1998.

[13] D. Slepian and H. O. Pollak, "Prolate spheriodal wave functions Fourier

analysis and uncertartainty-I," Bell Syst. Tech. J., vol. 40, pp. 43 - 64, 1961.

[14] R. W. Wies, J. W. Pierre, and D. J. Trudnowski, "Use of least mean squares

(LMS) adaptive filtering technique for estimating low-frequency

electromechanical modes in power systems," Proc. IEEE Power Eng. Soc.

Gen. Meet., vol. 2, pp. 1863 - 1870, Jun. 2004.


81




[16] R. A. Wiltshire, "Summary of PhD confirmation of candidature report: The





Australia, 2007.





'05, Nov. 2005.






damping in power systems," 7th IASTED Int. Conf. Sig. and Image Proc.,

Aug. 2005.

[22] D. J. Thomson, "Spectrum estimation and harmonic analysis," Proc. IEEE,

vol. 70, no. 1, pp. 1055 - 1096, Sep. 1982.

[23] P. O'Shea, "Detection and estimation methods for non-stationary signals,"

PhD Dissertation, University of Queensland, Brisbane, Australia, 1991.


IEEE Int. Conf. Acoust., Speech, and Signal Proc. (ICASSP), vol. 6, pp. 3570

- 3573, Jun. 2000.

Chapter 4: Damping Estimation Via

Spectral Averaging

4.1 Introduction It has been found empirically by Ledwich and Palmer [1] that the power system

output signal, , which occurs due to random load change disturbances, can be

well modeled as the output of a the power system IIR filter driven by integrated

white noise [2-8]. This is illustrated graphically in the left half of

( )ny

Figure 4.1. Also

illustrated on the right hand side of Figure 4.1 is the fact that, if one differentiates the

signal, , then the resulting signal is equivalent to the output of the IIR filter

driven by white noise. The model on the right hand side of

( )ny

Figure 4.1 is a very

simple model, and is the one used throughout this thesis for estimating damping from

power system in normal operation. So accordingly, when power system disturbance

records are obtained, they are first differentiated to obtain the signal, . This is

the signal which is used for subsequent parameter estimation.

( )nx

Figure 4.1 Power system model during post disturbance oscillations [5, 9].

Power system IIR filter

Differentiator

Integrator

WGN

( )nx

≡ Power system IIR filter

( )nx

WGN

Power system response, ( )ny

Chapter 4: Damping Estimation Via Spectral Averaging

83

Given that ( )nx can be modeled as the output of an IIR filter driven by white

Gaussian noise, the autocorrelation function of ( )nx will in theory be deterministic,

and will be a scaled version of the impulse response of the IIR filter. This ideal

autocorrelation function, however, can only be formed if a perfect ensemble

averaging of an infinite number of realizations can be performed. In practice, one

only has one realization, and so a time averaging must be performed to estimate the

autocorrelation function [10]. The autocorrelation function estimate formed via a

time average has two components, a deterministic one and a random one. The

deterministic component will be a scaled version of the impulse response of the IIR

filter and it will contain modal oscillations corresponding to the resonances in the IIR

filter. The random component will be additive coloured noise and the spectrum of

the colored noise will have the same shape as the frequency response of the IIR filter.

That is, the noise in the autocorrelation function estimate will have the same

resonances as the power system [2-8]. Because the noise spectrum has the same

shape as the deterministic spectrum, the signal to noise ratio (SNR) is constant across

the entire frequency band. This implies that Fourier based methods (such as sliding

window methods) could usefully employ information from many frequency positions

(bins) rather than just one, as they often do. This chapter investigates this possibility.

That is, damping is estimated using sliding spectral windows in which information

from many frequency bins is averaged.

As already mentioned, existing sliding window methods calculate the damping factor

based on information at only one frequency position [11-17]. The work in this

chapter uses damping estimates from different frequency bins and obtains an overall

estimate by averaging [9]. Simulations will be given which indicate that the sliding

window method implemented with Fourier spectral averaging can have a lower

damping estimate mean square error (MSE) than either basic sliding window or

Prony algorithms. The spectral averaging sliding window algorithms will also be

tested on some real power system data.


84

4.2 Spectral Averaging Methods 4.2.1 Sliding Window Spectral Averaging Methods For the basic sliding window method the acquired signal is processed by forming two

consecutive windows and taking the Discrete Fourier Transform (DFT) of each

window. In the traditional sliding window approach one uses the ratio of the

amplitudes in the two windows at the frequency of modal oscillation to determine the

damping factor. As mentioned in the previous section, however, one can obtain

damping estimates from all of the frequencies in the vicinity of the oscillation

frequency – the information at every different frequency bin is independent and the

signal to noise ratio is constant for all bins (assuming the model in Figure 4.1).

One of the critical issues to address is how to combine the information at different

frequency bins. Some form of averaging needs to be performed. The various

different possibilities are discussed below. For all methods it is assumed that as with

the sliding window methods in [11-14, 16, 17], the input signal is windowed with

two windows of length, , with each window separated by samples,

FFT of 1

wN gN

( ) =kZr1st window, FFT of 2( ) =kZr 2

nd window, =sT Sampling interval and

( )kω is the frequency sample in the thk −wN point DFT.

They are presented below:

1. Average spectral estimate sliding window method:

( )( ) ( )∑

−

=−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

=1

0 2

11 Relog11ˆ

w

sg

N

kTNkj

r

r

sgw ekZkZ

TNN ωα (4.2)

2. Average FFT ratio sliding window method:

( )( ) ( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

= ∑−

=−

1

0 2

12 Re1log1ˆ

w

sg

N

kTNkj

r

r

wsg ekZkZ

NTN ωα (4.3)


85

3. Average FFT sample sliding window method:

( )

( ) ( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

∑

∑−

=

−

−

=1

02

1

01

3 Relog1ˆw

sg

w

N

k

TNkjr

N

kr

sg ekZ

kZ

TN ωα (4.4)

The amplitude and other parameters can be estimated for each damping estimate via

methods presented in chapter 3 and the average value can be computed. The methods

can be adapted to multiple mode signals by using only the FFT samples in the

vicinity of the mode of interest.

Simulations of the various different estimators above are presented in Section 4.3. As

will be seen in that section, the average FFT ratio method performs better than any

alternative methods.

4.2.2 High Resolution Sliding Window Spectral Averaging Methods Parameters of closely space modes can be estimated using the same approach that

was used in [15, 18-23]. That is, the outputs of windowed (filtered) and down-shifted

signals can be fed into Prony algorithms [20, 21] (or variants of Prony’s method such

as the Kumaresan-Tufts method [22, 24]). The steps for high resolution methods with

spectral estimate averaging are:

Step 1: Compute FFT of each window:

( ) ( )∑−

=

−+=1

0,

wk

N

m

mjrd enmznkF ω (4.5)

where , , gvNn = Vv ,,2,1,0 K= =V number of windows.


86

Step 2: Downshift the resulting sequence by FFT frequency samples:

( ) ( ) ( )∑−

=

+−− +=1

0,

wkk

N

m

nmjr

njd enmzenkF ωω (4.6)

Step 3: Apply the resulting sequence to Kumaresan-Tufts method to determine

damping estimate,

thk

kα ; downshifted frequency estimate, thk k0ω and

downshifted complex amplitude estimate, .

thk

kdjkdeA φˆ

Step 4: Determine complex amplitude estimates by scaling the downshifted complex

amplitude estimates.

Step 5: Find the average of all estimates.

As with the non-high resolution methods discussed in Section 2.1, one has a number

of options to incorporate information from multiple frequency bins. These three

different possibilities (average spectral estimate, average FFT sample 1 and average

FFT sample 2) are illustrated in Figure 4.2.

FFT of multiple sliding

windows

Prony methods

Mean of estimates

Mean of FFT samples

Damping average FFT sample 2

Damping average spectral estimate

Prony methods

Mean of FFT samples

Damping average FFT sample 1

Prony methods

Demodulation to baseband

Demodulation to baseband

Figure 4.2 High resolution sliding window spectral averaging methods.


87

4.3 Simulations MATLAB software was used for testing of the algorithms presented in the previous

sections. The mean and median of the multiple damping estimates, multiple FFT

ratios and multiple FFT samples was determined. The lowest damping MSEs of the

two selection methods was plotted in Figures 4.3-4.6. In most cases the mean and

median provided the lowest MSE based estimates.

A single mode signal was generated and added to coloured noise. The following

simulated parameters were used for all simulations:

• Mode: signal length, 512=N ; sampling frequency, Hz; frequency, 4=sf

6.28320 =ω rad/s;

• Non-high resolution sliding window methods: number of samples between

windows and number of samples in a window, 256== wg NN ; number of FFT

samples used for averaging = 256;

• Prony methods: Linear prediction model order, 256=L ;

• High resolution sliding window methods: number of samples in a window,

; Step time, 128=wN 16=gN ; number of FFT samples used for averaging =

128;

• Number of simulation runs for each SNR level was 300.

In the first set of simulations, the non-high resolution method in Section 4.2.1 was

tested. The SNR was varied from 26 to 126 dB. Damping was, 02.0=α ; phase,

008.1=φ rad. The resulting MSEs are plotted in Figure 4.3. The average FFT ratio

sliding window method specified in (4.3) gave the best results.


88

30 40 50 60 70 80 90 100 110 120-200

-180

-160

-140

-120

-100

-80

-60

SNR (dB)

Dam

ping

est

imat

es M

SE (d

B)

Figure 4.3 Damping MSE vs SNR for basic sliding window (dots), average spectral estimate sliding

window and average FFT ratio sliding window (circles), average FFT sample sliding window

(squares); Kumaresan-Tufts Prony (full line) methods.

In the second set of simulations, the high resolution method in Section 4.2.2 was

tested. The SNR was again varied from 26 to 126 dB. Damping was, 02.0=α ;

phase, 008.1=φ rad. The resulting MSEs are plotted in Figure 4.4. Here the average

FFT sample sliding window methods provided lowest MSEs.


89

30 40 50 60 70 80 90 100 110 120

-170

-160

-150

-140

-130

-120

-110

-100

-90

-80

-70

SNR (dB)

Dam

ping

est

imat

es M

SE (d

B)

Figure 4.4 Damping MSE vs SNR for high resolution basic sliding window (dots), high resolution

average spectral estimate sliding window (circles), high resolution average FFT sample 1 and 2

sliding window (squares); Kumaresan-Tufts Prony (full line) methods.

4.4 Real Data Analysis Power system data was acquired in early 2005 from the Tasmanian power system

grid at a sampling rate of 10 Hz. The data is angle difference between Georgetown

and Creek Rd (Hobart) substations. Oscillating modal components were extracted

from real data via the formation of the autocorrelation function (See Figure 4.5). The

number of samples used was 50=N .


90

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10-10

Time (seconds)

Rea

l com

pone

nt o

f obs

erve

d si

gnal

, z r(t)

Figure 4.5 Autocorrelation function of disturbance signal from Tasmanian grid [9].

-5 -4 -3 -2 -1 0 1 2 3 4

2

3

4

5

6

7

8

9

10x 10-10

Frequency (Hz)

Four

ier t

rans

form

man

gitu

de o

f obs

erve

d si

gnal

, |Z

r(f)|

Figure 4.6 Fourier Transform magnitude of the autocorrelation function in Figure 4.5 [9].


91

The real data damping estimates and relative residual power between estimated and

acquired signals obtained for the various methods are shown in Tables 4.1 and 4.2:

Algorithm Damping estimate

Basic sliding window 0.58833

Averaging spectral estimates sliding window 0.62846

Averaging FFT ratio sliding window 0.58464

Averaging FFT sample sliding window 0.64148

Basic Prony 0.71535

Kumaresan-Tufts Prony 0.82397

High resolution basic sliding window 0.62726

High resolution averaging spectral estimates sliding window 0.56664

High resolution averaging FFT sample 1 sliding window 0.67208

High resolution averaging FFT sample 2 sliding window 0.65429 Table 4.1 Damping estimates.

Algorithm Residual (dB)

Basic sliding window -10.5553

Averaging spectral estimates sliding window -10.6257

Averaging FFT ratio sliding window -10.5421

Averaging FFT sample sliding window -10.6525

Basic Prony -10.4028

Kumaresan-Tufts Prony -9.6943

High resolution basic sliding window -10.6182

High resolution averaging spectral estimates sliding window -10.4415

High resolution averaging FFT sample 1 sliding window -10.6217

High resolution averaging FFT sample 2 sliding window -10.6042 Table 4.2 Residual between acquired and estimated signals.

It is seen that the high resolution average FFT sample sliding window method

provides the accurate with the smallest residual error power.


92

4.5 Conclusion Sliding window methods implemented with spectral averaging techniques can exploit

the colored nature of the noise typically found in large power systems. For this

reason it can outperform existing Fourier methods for damping factor estimation (at

least in some scenarios).

4.6 References [1] G. Ledwich and E. Palmer, "Modal estimates from normal operation of power


Jan. 2000.




[3] R. A. Wiltshire, "Summary of: PhD confirmation of candidature report - The





Australia, 2007.





'05, Nov. 2005.







93

damping in power systems," 7th IASTED Int. Conf. Sig. and Image Proc.,

Aug. 2005.



[10] S. M. Kay, Modern spectral estimation: theory and application. Englewood

Cliffs, N.J.: Prentice-Hall, 1988.

[11] K. K.-P. Poon and K.-C. Lee, "Analysis of transient stability swings in large





vol. 5, no. 1, pp. 148 - 153, Feb. 1990.






4, pp. 1261 - 1267, Nov. 2000.

[15] P. O'Shea, "A high-resolution spectral analysis algorithm for power-system

disturbance monitoring," IEEE Trans. Power Syst., vol. 17, no. 3, pp. 676 -

680, Aug. 2002.



Aug. 2005.



Aug. 2007.



5, pp. 2968 - 2972, Jun. 1998.


94



1, pp. 226 - 231, Feb. 1999.









IEEE Int. Conf. Acoust., Speech, and Sig. Proc. (ICASSP), vol. 6, pp. 3570 -

3573, Jun. 2000.




95

Chapter 5: Weighted Least-Squares

Averaging Method for Damping

Estimation in Power Systems

5.1 Introduction Sliding window [1-7] and Prony [8-13] methods are not optimal estimation

methods. They give rise to MSEs which do not meet the Cramer-Rao bound

(CRB). This chapter applies optimal estimation techniques to estimating damping

in power systems. While these optimal estimation techniques are not new, their use

in power system anlysis is new. Within this chapter comparisons are made with

existing methods such as Fourier based sliding window and Prony algorithms.

5.2 Optimal Estimation Theory Averaging Based Methods 5.2.1 Signal Model It is assumed initially that there is only one modal component present in the

observation. This observed signal model is defined as:

( ) ( ) ( )nnznz sr ε+= , 1,,1,0 −= Nn K (5.1)

( ) ( )nAe nj εφω += +0 (5.2)

where =A Amplitude, =α Damping, =0ω Modal (angular) frequency, =φ Initial

phase and ( ) =nε additive white complex Gaussian noise of variance . A

sampling rate of 1 samples/second is assumed.

2σ

Note that the signal is assumed to be complex. Of course, the processed real data

consists of only real values. A complex “analytic signal” can, however, be obtained

Chapter 5: Weighted Least-Squares Averaging Method for Damping Estimation in

Power Systems

96

from real data by taking the Fourier transform, setting the negative frequency

components to zero and then taking the inverse Fourier transform.

5.2.2 Least Squares Estimators A sequence of preliminary damping estimates can be computed by taking

the ratio of two data sequences separated by one sample and then performing some

very simple processing:

1−N

( ) ( ) ( )( )[ ]nznzny rrr /1logReˆ −= 1,,2,1, = −Nn K (5.3)

If the additive noise power is zero then (5.3) will simplify to:

( ) ( )[ ]αω+−= jr eny logReˆ (5.4)

[ ]αω +−= jRe (5.5)

α= , 1,,2,1 −= Nn K (5.6)

It can be seen from the above that in the absence of noise, (5.3) yields the modal

damping factor of the modal oscillation. The algorithms presented in his chapter

use this relationship in (5.3) to compute the modal damping. A number of different

damping estimation algorithms are described below.

5.2.2.1 Estimator 1

One can obtain a sequence of preliminary estimates of damping by evaluating (5.3)

at many different values of n. Figure 5.1 shows a typical sequence of preliminary

damping estimates. The estimates shown in Figure 5.1 are very noisy, and some

processing is needed to obtain a final damping estimate. For Estimator 1 the final

97

damping estimate is obtained as the average of all these preliminary estimates:

( ) ( )( )[∑−

=

−=1

11 /1logRe1ˆ

N

nrr nznz

Nα ] (5.7)

The above estimator is the least-squares estimator of damping, based on all the

preliminary estimates from (5.3). Estimator 1 is very simple but it is not an using

the concept of optimal estimator. Simulations presented in Section 5.3 will verify

this fact. That is, damping estimate MSEs for Estimator 1 do not meet the CR

bound.

The optimal estimator would need to account for the colour of the noise present on

the preliminary estimates. The optimal estimator is a weighted least-squares

estimator. It is presented below.

0 50 100 150 200 250 300 350 400-15

-10

-5

0

5

x 10-3

Sample number

Sequ

ence

of e

stim

ates

, y r(n

)

Figure 5.1 Sequence of estimates, ( )nyr for 0043.0−=α .

The true and preliminary damping estimates can be expressed in the following

matrix equation:


Power Systems

98

ryW =+ eXα (5.8)

where ,( ) ( )[ ]Trrr Nyy 1ˆ1ˆ −= Ky [ ]TX 111 K= ,

( ) ( ) ( )[ ]Te Nwww 121 −= KW and ( ) ( ) α−= nynw r . is the additive noise of

damping estimate sequence . Assuming the linear signal model in (5.8) and

Gaussian noise in , the damping can be optimally estimated with weighted least

squares methods. This is the basis of Estimator 2 described below.

eW

ry

eW

5.2.2.2 Estimator 2

The weighted least squares damping estimate is:

( )[ ] ( ) ( )([ ]∑−

=

−−− −=1

1

1112 /1logRe1ˆ

N

nrr

TT nznzXXXN

CCα ) (5.9)

where C is the Covariance matrix. By defining the weighted window as:

( ) ( )[ ]111 −−−= CC TT XXXnv (5.10)

the damping estimate can be represented as:

( ) ( ) ( )([∑−

=

−=1

12 /1logRe1ˆ

N

nrr nznznv

Nα )] (5.11)

99

The covariance matrix, C, is defined by:

( ) ( ) jwiwECij*= (5.12)

( )

( )( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=−−

=

= −−

−−

otherwise 0

1for

for 2

1,max22

2

122

2

jieA

jieA

ji

i

α

α

σ

σ

(5.13)

To determine the C matrix one needs to know the damping factor which is not

known a priori. One can, however, use an initial damping value estimate that can

be determined from other algorithms. The simplest estimate is obtained by finding

how long it takes the mode to decay to 1/10 of its original value (from visual

inspection). Then one can infer an estimate of damping rate.

Simulations presented in Section 5.3 will show that Estimator 2 gives damping

estimates which are optimal above a given SNR threshold. That is, the damping

estimates meet the CR bound above a given threshold.

5.2.3 Low SNR Operation While Estimator 2 is optimal above a given SNR threshold, it is not effective at

low SNR values and it is not effective for processing signals containing multiple

modes. Estimator 3, which is described below, allows operation at low SNR. The

rational is that one first estimates the frequency of the mode. Then one shifts the

modal component down to baseband and then does a filtering which reduces the

noise. The downshifting and filtering process does not change the damping, even

though it does reduce the noise. One can therefore do the damping estimation on a

signal which has had its SNR enhanced.

5.2.3.1 Estimator 3

First consider a single mode. Estimator 3 is implemented according to the

following procedure.


Power Systems

100

Step 1: Find the modal frequency using the Fourier transform:

( )⎭⎬⎫

⎩⎨⎧

= ∑−

=

−1

00

0

0maxargˆ

N

n

njr enz ω

ωω (5.14)

Step 2: Downshift the observed signal to baseband:

( ) ( ) njrrd enznz 0ω−= (5.15)

Step 3: Apply a moving average filter to reduce the noise:

( ) ( ) ( )nhnznz rdf *= (5.16)

where:

( )⎩⎨⎧ =

=otherwise0

,,11 Mnnh

K

Step 4: Determine the damping from the following:

( ) ( )∑−

=

=1

13 ˆ1ˆ

N

nff nynv

Nα (5.17)

( ) ( ) ( )([∑−

=

−=1

1

/1logRe1 N

nfff nznznv

N)] (5.18)

where is the covariance matrix of fC ( )ny f .

101

This covariance matrix is:

( ) ( )

( )

⎪⎩

⎪⎨

⎧

=−=+

≈ −+

+−−

otherwise 0for e-

for 2

121- -2

ijf Mjijiee

ji

Mii

α

αα

C (5.19)

Simulations in the next section will show that the above procedure yields results

which are optimal at lower SNRs than for Estimator 2.

For multiple modes, one can still use the above method as long as the modes are

well separated in the frequency domain. This is so because the filtering process in

(5.16) of the above algorithm filters out not only the noise but also the other

components. One can therefore process each of the components in turn.

5.3 Simulations MATLAB software was used for testing of the algorithms in the previous sections.

In the first set of simulations, Estimators 1, 2 and Prony’s method were used to

estimate damping for a signal with the following parameters: amplitude, =A 1;

damping, =α –0.0025; frequency, =0ω 0.2319 rad/s; signal length, 512

samples. The signal was immersed in additive white Gaussian noise. The mean

square error (MSE) in damping was plotted against SNR. 100 Monte-Carlo

simulations were run and the results are shown in

=N

Figure 5.2.


Power Systems

102

0 5 10 15 20 25 30 35 40-120

-100

-80

-60

-40

-20

0

20

Mea

n sq

uare

erro

r in

dam

ping

(dB

)

SNR (dB)

Figure 5.2 Damping MSE vs SNR for Estimator 1 (cross signs), Estimator 2 (circles) and Prony

method (plus signs). Cramer-Rao bound is denoted by dashed line.

It can be seen that Estimator 2 (the optimal weighted least-squares estimator) is

very close to the Cramer-Rao bound, above a threshold of about 10 dB. This

supports the claim that Estimator 2 is optimal, at least above a threshold. The other

estimators all have MSEs which are above the Cramer-Rao bound. i.e. they are not

optimal.

Figure 5.3 shows the results of simulations for the low SNR algorithm in Section

5.2.3 (i.e. for Estimator 3). The MSEs for the “Modified Estimator 1” and the

“Modified Prony method” are also shown in Figure 5.3. The Modified Estimator 1

algorithm consists of the first three steps of the algorithm in Section 5.2.3 followed

by the application of Estimator 1. The Modified Prony method consists of the first

three steps of the algorithm in Section 5.2.3 followed by the application of the

standard Prony estimator. It is clear from Figure 5.3 that Estimator 3 (the Optimal

103

Estimation Theory based estimator) still performs the best. It is also seen that the

SNR enhancing procedure in Steps 1-3 of Section 5.2.3 helps to improve the

performance of all three methods simulated.

-10 -5 0 5 10 15 20 25 30-100

-90

-80

-70

-60

-50

-40

-30

Mea

n sq

uare

erro

r in

dam

ping

(dB

)

SNR (dB)

Figure 5.3 Damping MSE vs SNR for Estimator 3 (circles), Modified Estimator 1 (cross signs) and

Kumaresan-Tufts Prony method (plus signs) for simulated single mode data. Cramer-Rao bound is

denoted by dashed line.

5.4 Real Data Analysis 5.4.1 Processing of power system signals The assumption at the beginning of this chapter was that the observation is

embedded in white Gaussian noise. In practice the noise will probably not be

white. According to the findings in [14], the observation is likely to have a

significant coloured noise component in addition to a small white noise

component. The coloured noise component will tend to have the same resonances

as the power system itself. It is important to comment on how these algorithms will

perform when the noise is not white as assumed. This is addressed in what follows.


Power Systems

104

Normally power system signals have multiple components and so one should use

the 4-step algorithm in Section 5.2.3 to process each of the modes in turn. If this

algorithm is used, the requirement that the noise be white effectively relaxes to a

requirement that the noise be flat in the vicinity of the modes of interest. The

reason for this relaxation is that the noise outside the pass-band of the filter in

(5.16) is largely removed; it is therefore irrelevant what the noise is outside of this

band. As a consequence of this relaxation the weighted least-squares algorithm still

produces quite good estimates even when the noise on the observation is coloured

(provided that there is good frequency separation of modal components). This in

turn means that the algorithm is suited to use in a power systems scenario.

Multi-site generator angle vibration measurement data was acquired from

Adelaide, Brisbane, Melbourne and Sydney monitoring stations during 10th April

2004 event [15]. The angle data contains less noise than voltage, current or power

flow data. The noise is sometimes caused by local load variations. The observed

signal from Melbourne monitoring station is shown in Figure 5.4 and its FT

magnitude is shown in Figure 5.5.

105

0 5 10 15 20 25 30 35 40 45-30

-20

-10

0

10

20

30

Time (seconds)

Rea

l com

pone

nt o

f obs

erve

d si

gnal

, Re[

z r(t)

]

Figure 5.4 Acquired real data.

0 0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

2.5

x 104

Frequency (Hz)

Obs

erve

d si

gnal

Fou

rier t

rans

form

mag

nitu

de, |

Z r(f

)|

Figure 5.5 Fourier transform magnitude of acquired real data.


Power Systems

106

Input sequence was estimated with estimates of signal model parameters. Relative

residual power between acquired and estimated sequences was calculated by

various methods and the results are shown in Table 5.1:

Algorithm Residual (dB)

Estimator 2 -18.7529

Estimator 3 -21.2081

Kumaresan-Tufts Prony -20.9707 Table 5.1 Residual between acquired and estimated signals.

5.5 Discussion The methods presented in this paper are very computationally efficient. Estimators

1 and 2 require only computations, while Estimator 3 requires ( )NO ( )2NO

computations. By comparison Prony’s method typically requires ( )3NO

computations.

Although Estimator 3 is computationally more complex than Estimator 2, it has the

advantage of being able to work at low SNRs (due in large part to the filtering

stage in the estimation process).

Estimator 3 can be used effectively for practical power system signals, provided

that there is good separation of modes. This has been confirmed by the real data

analysis example in this paper. If some modes in a real data example are not well

separated, then it is best to use an alternative method (such as the Kumaresan-Tufts

technique or High Resolution Multiple Sliding Window methods in Chapter 3) for

analyzing those modes.

107

5.6 Conclusion Simulations and real data analysis show that lower damping MSE can be obtained

with the use of weighted least squares averaging damping estimation methods

compared with the basic sliding window or Prony methods.

5.7 References [1] K. K.-P. Poon and K.-C. Lee, "Analysis of transient stability swings in

large interconnected power systems by Fourier transformation," IEEE



with heat phenomenon by Fourier transformation," IEEE Trans. Power

Syst., vol. 5, no. 1, pp. 148 - 153, Feb. 1990.





power system disturbance monitoring," IEEE Trans. Power Syst., vol. 15,

no. 4, pp. 1261 - 1267, Nov. 2000.

[5] M. Glickman and P. O'Shea, "Damping estimation of electric disturbances

in distributed power systems," 7th IASTED Int. Conf. Sig. and Image Proc.,

Aug. 2005.


interconnected power systems," IEEE Region 10 - TENCON '05, Nov.

2005.

[7] M. Glickman, P. O'Shea, and G. Ledwich, "Estimation of modal damping

in power networks," IEEE Trans. Power Syst., vol. 22, no. 3, pp. 1340 -

1350, Aug. 2007.





Power Systems

108


identification from measured ringdowns," Proc. American Control Conf.,

vol. 5, pp. 2968 - 2972, Jun. 1998.


more accurate using multiple signals," IEEE Trans. Power Syst., vol. 14,

no. 1, pp. 226 - 231, Feb. 1999.








[14] G. Ledwich and E. Palmer, "Modal estimates from normal operation of

power systems," Proc. IEEE Power Eng. Soc. Wint. Meet., vol. 2, pp. 1527

- 1531, Jan. 2000.

[15] G. Ledwich and C. Zhang, "Disturbance Report (unpublished work),"

Unpublished, Queensland University of Technology, Brisbane, Australia,

Apr. 2004.

Chapter 6: General Discussion

6.1 Summary The purpose of the research study was to devise improved ways of determining

damping in power systems. Several new techniques were devised and applied to

determining modal damping from observations of both real and simulated power

systems.

A new approach based on the use of multiple orthogonal sliding windows was

proposed in Chapter 3. The rationale behind using these orthogonal windows was

that with orthogonal windows one could obtain independent sets of data which could

then be averaged to yield better estimates. Additionally, because the approach was

based on Fourier analysis, it was anticipated that the performance would be good at

relatively low SNR. A statistical analysis of the new approach was performed, as

were simulation studies [1]. Simulations and real data analysis confirmed that one

could obtain better estimates (as expected) under a number of circumstances. It was

found, however, that the new approach did not perform better in all circumstances. In

particular, it was found the new methods were best suited to analyzing modes which

did not have very short duration. The findings emphasized the fact that one should

use a suite of tools for analyzing modes in a realistic power system scenario. In such

scenarios, where one does not know the true damping factors, the notion of a

“residual power” has been found to be a useful measure of performance [2]. This

residual energy is essentially the energy of the difference between the observed

signal and the signal reconstructed from parameter estimates.

One of the particularly interesting features of the new methods based on orthogonal

windows is that one has access to several different estimates corresponding to the

different windows. This provides a means for cross-validation of the results and

therefore gives an indication of the reliability of estimation.


110

In Chapter 4 the issue of noise on power systems in normal operation was addressed.

Based on the empirical model of Ledwich and Palmer [3] the noise on the

observation will not be white but will have the same spectral character as the true

modes. The methods introduced in Chapter 4 took the character of the noise into

account when performing the estimation. In particular, they took advantage of the

fact that the SNR was constant across all frequencies. That is, although the modal

signal strength varies in the windowed spectrum as a function of frequency, so does

the noise strength; moreover the noise and signal strength vary in such a way that the

SNR is constant. This enables multiple damping estimates to be formed from

multiple frequency positions in the sliding window spectra. The estimates obtained

from these various frequency positions can then be averaged. The simulations

undertaken in Chapter 4 showed that improved performance can be obtained by

using the new multi-sample spectral averaging methods (assuming of course that the

noise does indeed have the same spectral character as the true modes).

In Chapter 5, weighted averaging methods were applied to estimating the damping of

modal oscillations in power systems. This is, to the best of the author’s knowledge,

the first time optimal methods have been used for parameter estimation of power

system data. The optimal estimation methods use a simple pre-processing of the

observation along with weighted least-squares techniques to do the estimation. The

technique is suitable for noise of any colour, a fact which is fortuitous for power

system data. Both simulated and real data showed that this technique has potential

for improved accuracy in damping estimation. As well as giving very good statistical

performance, the optimal parameter estimation methods are very computationally

efficient.

Each algorithm has limits on the maximum values of damping that it can estimate.

High resolution sliding window methods use Prony techniques, thus their

performance is highly influenced by the type of Prony method that is used.


111

The methods have been tested on various types of data presented in the thesis and

published papers. However, the improved methods cannot outperform existing

algorithms for all simulation parameters and data types.

Increase in damping increases the steepness of the signal and reduces the magnitude

of sequence values at the end of the sampled sequence where the signal amplitude

decays to zero. Thus the estimation error increases due to low precision. Simulations

show that high resolution methods work better with high damping magnitude that

low resolution sliding window methods.

Increasing the window length increases precision only to a certain point until the

value reaches an optimum point when the MSE begins to fall. Reducing the damping

makes the signal look like a spike in frequency domain rather than a bell shaped

curve. This will reduce the precision of damping values estimated by spectral

averaging methods.

6.2 Research Outcomes A number of new methods have been proposed for estimating the damping of power

system modes and the conclusions from testing of each of the new methods are

presented below.

A new technique based on orthogonal multiple sliding window analysis has

outperformed existing sliding window methods and shows significant potential for

parameter estimation, particularly at low SNR [1, 2]. The multiple orthogonal

windows provide multiple independent estimates which can be used for cross

validation. While the orthogonal window method performs very well in various

situations, it does not always outperform the Kumaresan-Tufts method [4], and so

should be used as one of a number of possible analysis tools. When real data is being

analysed the residual power should be used to evaluate the most appropriate tool to

be used.


112

Another new method introduced was the spectral averaging sliding window

approach. This method exploited the fact that in practical power systems the noise

tends to have the same spectral character as the signal modes. Accordingly, the SNR

is approximately constant for all frequency samples in the spectrum. The new

method formed multiple sliding window based damping estimates from multiple

frequency samples and then averaged them. The performance of the new method was

found in simulations to be very good compared to traditional sliding window

methods (providing the noise had the same spectral shape as the true modes) [5].

The thesis also applied optimal estimation methods to estimating modal damping

from power system signals. The performance was seen to be optimal above an SNR

threshold, provided that the assumed signal model was appropriate. Furthermore the

computational efficiency of the method was good, which augers well for real-time

implementation. The method was also found to be effective for a real power system

example.

The methods devised in this thesis have potential application in real power systems.

They can be used generally for determining damping levels in power system output

data. Some specific applications could be the testing of new procedures to control

damping in power systems.

6.3 Future Work One of the most promising avenues for future investigations would be extension of

the optimal estimation technique of Chapter 5 to be able to process closely spaced

modes. Currently, the method is only applicable to modes that are clearly separated

in the spectrum. The traditional means for estimating damping for closely spaced

modes is the Prony based Kumaresan-Tufts method [4], but this method is

computationally intensive and is not in general optimal. It would be challenging to

find a computationally simple method for optimally estimating damping in closely

space modes. Nonetheless, optimal and practically realizable methods should be the


113

goal of all engineering fields, and so the development of such methods for closely

spaced modes would be a very worthwhile path to pursue.

In addition to modifying the optimal methods to be able to process closely spaced

modes, it would be useful to modify them to be effective at lower SNR thresholds.

The orthogonal filtering functions discussed in Chapter 3 may be useful in this

regard.

6.4 References [1] M. Glickman and P. O'Shea, "Damping estimation of electric disturbances in


Aug. 2005.



Aug. 2007.



Jan. 2000.






Chapter 7: Conclusion

This thesis has really addressed the important issue of whether or not the accuracy of

damping estimates in large distributed power systems can be improved. This is a

critical issue because over the past decade or so power systems have come

increasingly interconnected and therefore increasingly vulnerable. If a generator fails

in a modern power system it can trigger a domino effect of outages in other

generators, with large regions being subsequently blacked out. The New York state

blackout in 2003 was a prime example. The finding of this thesis has been that

accuracy of damping estimates can be improved, and in practically useful ways.

The use of multiple orthogonal sliding windows has been found to give accuracy

improvements in a number of different power system scenarios. The orthogonal

window method has been found to have a particularly appealing feature – it gives a

series of independent estimates which can be used for cross-validation purposes.

Additionally, if used in conjunction with spectrogram/demodulation/filtering and

sample reduction techniques, the method can be realised in a computationally

efficient manner. Computational efficiency is important if one is to correct dangerous

damping levels without large time delays.

A key finding of the thesis is that one can not only improve the accuracy of damping

estimates, but that one actually find optimal estimates of damping, provided some

conditions hold. These conditions include a requirement that the oscillating modes

being analysed are well separated in the spectral domain and that the SNR in the

observation is above a threshold. The SNR threshold requirement can usually be met

in practice but the spectral separation requirement is not met for all power system

scenarios. Currently, one would only be able to use the optimal estimation methods

for those modes which are well separated; extension to closely spaced modes is an

area for future work. It has also been found that the optimal damping estimation

Chapter 7: Conclusion

115

methods are very computationally efficient and so lend themselves well to flagging

damping anomalies quickly.

Appendix

8.1 Damping Factor Estimation from Two Sliding Windows The derivations in this section are based on derivations presented in [1, 2]. Assuming

that the single mode in power system defined in equation 3.1 is sampled with two

windowed signals (3.3) and (3.4) the ratio of Fourier transform of the two windowed

signals is given by:

( )( )

( )( )⎟

⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛

0

0

0

0

2

1

2

1

ωω

ωω

s

s

r

r

ZZ

ZZ

(8.1)

( ) ( )

( )( ) ( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

∑

∑−

=

−+++−

−

=

−++−

1

0

1

0

00

00

wg

w

N

n

njjNnj

N

n

njjnj

enwAe

enwAe

ωφωα

ωφωα

(8.2)

( ) gNjae 0ω−= (8.3)

Therefore the damping factor is:

( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡= − gNj

s

s

g eZZ

N 0

2

1

0

0ln1ωω

ωα (8.4)

The amplitude and phase are:

( )w

sj

SZ

Ae 01ωφ = (8.5)

where:

( )∑−

=

−=1

0

wN

n

nw enwS α (8.6)

116

Appendix

In practice, the observed signal will have nonzero noise power. Therefore:

( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

= − gNjr

r

g eZZ

N 0

2

1

0

0Reln1ˆ ωωω

α (8.7)

( )( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+

+= − gNj

s

s

g eqZqZ

N 0

2

1

20

10Reln1ωω

ω (8.8)

where:

( ) ( )∑−

=

−=1

01

0wN

n

njenwnq ωε (8.9)

and

( ) ( )∑−

=

−+=1

02

0wN

n

njg enwNnq ωε (8.10)

Therefore:

( )( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

++

= −++− gg Njw

jNjw

j

g eqSAeqSAe

N 002

1Reln1ˆ ωφωα

φ

α (8.11)

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

++

= −+− gg

g

Njw

jNw

jN

g eqSAeqSAee

N 02

1Reln1ωφα

φα (8.12)

( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+

+= −+−

−

1

2

11

01

1Reln1gg

g

NjNw

j

wj

N

g eSAeq

SAeqeN ωαφ

φα (8.13)

Assuming that and are small compared with 1q 2q ( )01ωsY and ( 02

)ωsY the following

approximations can be used ( ) xx −≈+ 11/1 . Therefore:

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+≈ +− g

g

Njw

jw

jN

g eSAeq

SAeqe

N 0

21 11Reln1ˆ ωαφφαα (8.14)

117

( ) ( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−−+≈

−−

2222121

00

1Reln1

wj

Nj

wj

Nj

wj

N

g SeAeqq

SAeeq

SAeqe

N

ggg

φ

ωα

φ

ωα

φα (8.15)

( ) xx ≈+1ln . Therefore:

( ) ( )

⎭⎬⎫

⎩⎨⎧

−−+≈−−

2222121

00

Re1ˆw

j

Nj

wj

Nj

wj

g SeAeqq

SAeeq

SAeq

N

gg

φ

ωα

φ

ωα

φαα (8.16)

( )( )

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−−+=

−−

wj

NjNj

wj

g SAeeqqeqq

SAeN

gg

φ

ωαωα

φα0

0 2121

1Re1 (8.17)

If the windows are not overlapping than covariance component equals to zero.

Therefore:

( ) ( )( )2

221

2varvar)ˆvar(

Wg

N

SANeqq g

⋅+

≈α

α (8.18)

( ) ( )( )⎥⎥⎦

⎤

⎢⎢⎣

⎡+= ∑

−

=

1

0

22222

2

12

wg

N

n

N

wg

nweSAN

ασ (8.19)

( ) ( )( )

( )21

0

1

0

22

22

21

2∑

∑−

=

−

−

=

+=

w

wg

N

n

n

N

n

N

g enw

nwe

ANα

α

σ (8.20)

8.1.1 References [1] M. Glickman, P. O'Shea, and G. Ledwich, "Estimation of modal damping in


Aug. 2007.

118

Appendix



4, pp. 1261 - 1267, Nov. 2000.

119

8.2 Matlab Code used to Generate Simulated Real Data in

Chapter 3 % close all % 4 machines J1=10; J2=16; J3=14; J4=4.5; x12=0.015; x23=0.018; x34=0.02; randn('seed',sum(100*clock)) A=[0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 -1/(J1*x12) 1/(J1*x12) 0 0 -3.6/J1 0 0 0 1/(J2*x12) (-1/(J2*x12)-1/(J2*x23)) 1/(J2*x23) 0 0 -2.2/J2 0 0 0 1/(J3*x23) -1/(J3*x23)-1/(J4*x34) 1/(J4*x34) 0 0 -2.4/J3 0 0 0 1/(J4*x34) -1/(J4*x34) 0 0 0 -12.4/J4 ]; B=[0 0 0 0;0 0 0 0;0 0 0 0;0 0 0 0;1/J1 0 0 0;0 1/J2 0 0;0 0 1/J3 0;0 0 0 1/J4]; C=[1/x12 -1/x12 0 0 0 0 0 0 0 1/x23 -1/x23 0 0 0 0 0]; D=zeros(2,4); K=0.0000*[0 0 0 1 1 1;0 0 0 1 1 1;0 0 0 1 1 1]; al=1.02*[-0.05 0 0;0 -0.05 0;0 0 -0.05]; ac=al; scale=500; JT=J1+J2+J3; b=0.001; h=-10; Atot=[A [zeros(4,1);h/J1; h/J2; h/J3;h/J4 ];zeros(1,4) b*J1/JT b*J2/JT b*J3/JT b*J4/JT -b]; Btot=[B;zeros(1,4)]; Ktot=[K zeros(3,3)]; Ctot=[C zeros(2,1)]; Dtot=D; [V,ev]=eig(Atot) f_hz=1/(2*pi)*diag(ev) dt=0.1; T=0:dt:10000; n=length(T); u1=randn(1,n); u2=randn(1,n); u3=randn(1,n); u4=randn(1,n); ui1=filter(1,[1 -.9999],10*u1); ui2=filter(1,[1 -.9999],10*u2); ui3=filter(1,[1 -.9999],10*u3); ui4=filter(1,[1 -.9999],10*u4);

120

Appendix

U=[ui1'/J1 ui2'/J2 ui3'/J3 ui4'/J4]/100; [Y,X]=lsim( Atot,Btot,Ctot,Dtot,U,T); figure(1) plot(T,Y,T,X) xlabel('Time') dtt=T(2)-T(1); tt=0:dtt:20; figure(2) [Yi,Xi,Tvi]=impulse(Atot,Btot,Ctot,Dtot,1,tt); plot(Tvi,Xi) xlabel('Time') coi=X(:,1:4)*[10;16;14;4.5]/44.5; coiv=X(:,5:8)*[10;16;14;4.5]/44.5; dv=diff(X(:,1:4)-1*coi*[1 1 1 1]); dvel=diff(X(:,5:8)-coiv*[1 1 1 1]); mm=600; t=(-mm:mm)*dt; ddc=diff(diff(coi)); ddc=dv(:,2); r1=xcorr(dv(:,1),mm); r1v=xcorr(dvel(:,1),mm); r1c=xcorr(dv(:,1),ddc,mm); r2c=xcorr(dv(:,2),ddc,mm); r3c=xcorr(dv(:,3),ddc,mm); r4c=xcorr(dv(:,4),ddc,mm); r1v=xcorr(dvel(:,1),mm); r1vc=xcorr(dvel(:,1),ddc,mm); r2vc=xcorr(dvel(:,2),ddc,mm); r3vc=xcorr(dvel(:,3),ddc,mm); r4vc=xcorr(dvel(:,4),ddc,mm); r2=xcorr(dv(:,2),mm); r3=xcorr(dv(:,3),mm); r4=xcorr(dv(:,4),mm); r2v=xcorr(dvel(:,2),mm); r3v=xcorr(dvel(:,3),mm); r4v=xcorr(dvel(:,4),mm); rc=xcorr(diff(diff(coi)),mm); figure(3) plot(t,r1,t,r2,t,r3,t,r4) title('Auto corr of load') legend('1','2','3','4') figure(4) tv=(0:length(dv)-1)*dt; plot(tv,dv(:,1),tv,dv(:,2),tv,dv(:,3)) title('diff of angles'); figure(5) tt=0:dt:mm*dt; tor=40; et=exp(-tt/tor); f1=fft(et'.*r1(mm+1:2*mm+1)); f2=fft(et'.*r2(mm+1:2*mm+1)); f3=fft(et'.*r3(mm+1:2*mm+1)); f4=fft(et'.*r4(mm+1:2*mm+1)); f1c=fft(et'.*r1c(mm+1:2*mm+1)); f2c=fft(et'.*r2c(mm+1:2*mm+1));

121

f3c=fft(et'.*r3c(mm+1:2*mm+1)); f4c=fft(et'.*r4c(mm+1:2*mm+1)); f1vc=fft(et'.*r1vc(mm+1:2*mm+1)); f2vc=fft(et'.*r2vc(mm+1:2*mm+1)); f3vc=fft(et'.*r3vc(mm+1:2*mm+1)); f4vc=fft(et'.*r4vc(mm+1:2*mm+1)); f1v=fft(et'.*r1v(mm+1:2*mm+1)); f2=fft(et'.*r2(mm+1:2*mm+1)); f3=fft(et'.*r3(mm+1:2*mm+1)); f4=fft(et'.*r4(mm+1:2*mm+1)); fc=fft(et'.*rc(mm+1:2*mm+1)); nf=length(f1); f=(0:nf-1)/(nf*dt); plot(f,abs(f1c),f,abs(f2c),f,abs(f3c),f,abs(f4c),f,abs(fc)) grid on title('fft') legend('1','2','3','4','coi') figure(6) plot(f,abs(f1vc),f,abs(f2vc),f,abs(f3vc),f,abs(f4vc)) grid on title('fft vel') legend('1','2','3','4')

122

Documents

Doctor of Philosophy Thesis - QUTeprints.qut.edu.au/16497/1/Mark_Glickman_Thesis.pdf · Doctor of Philosophy Thesis Disturbance Monitoring in Distributed Power Systems Mark Glickman,