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Title: The Hilbert Transform: Applications in the Analysis of Power Engineering Dynamics


1
The Hilbert Transform Applications in the
Analysis of Power Engineering Dynamics
Presentation at the Missouri University of
Science and Technology
  • G. T. Heydt
  • Arizona State University
  • October, 2009

2
It is obvious that
2??-?/??4
?
?5
?
  • µf21 2psgQbuuY(t) gtgt 12.1028746x(z)
  • p?? O?/0.24r n?5??? ?/?-1 \ 1.1111??/??v?
  • q34-atan(acos(z))?cosh(q)/?20.2957352957?q
  • 8?21/?-1 ?

?2-3?8gt1
??
Except when n is odd or n 3
57?q
VERY IMPORTANT FOR Z 2???
Note
2
3
Outline
  • Why look to transform theory for any help in
    power system dynamic analysis?
  • The Hilbert transform
  • Some interesting mathematics
  • Modal analysis, damping and stability
  • Some complications
  • Summary, conclusions, recommendations, possible
    venues for new work in power engineering

3
4
Objectives
  • To introduce the Hilbert transform in a
    comprehensible way
  • To discuss applications in power engineering
  • To give a capsule summary of challenges in the
    area

5
Il existe de nombreuses façons d'afficher une
image
transform
  • What is an image
  • A way to see something
  • A view not easily interpreted otherwise
  • Trans across Form manifestation

TRANSFORMATION
A mapping from one space to another
6
Il existe de nombreuses façons d'afficher une
image
  • The concept is to make calculations easier in the
    transformed domain
  • And not to waste too much time in transforming
    and untransforming

TRANSFORMATION
A mapping from one space to another
7
Issues in power signal identification
Transforms are often useful for these applications
Take corrective control action, alarms, PSS
signals
POWER SYSTEM
IDENTIFICATION
Measurements
  • The main contenders
  • Fourier analysis
  • Prony analysis
  • Hilbert analysis
  • Various control theory approaches such as
    observer design

8
Why use transformations?
Fourier transform Laplace transform Hartley
transform
To convert a differential equation to an
algebraic equation To convert the convolution
integral into something that is more easily
calculated To convert a signal with a wide
frequency bandwidth into something that has a
narrow bandwidth in the transformed domain To
get rid of unbalanced three phase quantities To
make calculations easier And to conform with
widely used notation
Fourier transform Laplace transform Hartley
transform Discrete Hartley and Fourier transforms
Walsh transform
Symmetrical components, Clarkes components
9
David Hilbert
  • 1862 1943
  • born in Königsberg, East Prussia
  • algebraic forms
  • algebraic number theory
  • foundations of geometry
  • Dirichlet's principle
  • calculus of variations
  • integral equations
  • theoretical physics and dynamics
  • foundations of mathematics
  • the Hilbert transform

10
The Hilbert transform
Some points of interest The transformed variable
is still t The convolution integral is best
performed by taking the FT of both sides and
use the convolution property of the FT Recall
that the FT of the 1/t term is jsgn(?) This can
be verified by the reciprocity theorem if f(t)
and F(j?) are transform pairs, then f(j?) and
F(t) are also transform pairs
11
The Hilbert transform
The FT of the 1/pt term is jsgn(?) This can be
verified by the reciprocity theorem if f(t) and
F(j?) are transform pairs, then f(j?) and F(t)
are also transform pairs
These are FT transform pairs
12
The Hilbert transform
Therefore, one way to obtain the HT is to
MULTIPLY the FT of 1/pt (namely sgn(?)) with the
FT of x(t).
But that is easy just reverse the signs of all
the terms of the FT of x(t) over negative values
of ?. Then take the IFT if you really need X(t).
13
Some rather interesting Hilbert transforms
x(t) X(t)
Special interest in dynamic studies of all kinds
of linear systems
14
Some rather interesting properties of the Hilbert
transform
Linearity
H(ax(t))aX(t) H(x(t)y(t))X(t)Y(t)
When the HT is applied twice to x(t), the result
is x(t). This is also called anti-involution.
Double application
H-1 -H
Inverse HT
Differentiation
H(dx/dt) dH(t)/dt
H(xy) Xy xY
Convolution
The analytic function
XA(t) x(t) j Hx(t)
15
The analytic function of the decaying sinusoid
HT pair
XA(t) x(t) j Hx(t)
The analytic function
Therefore XA(t) Aestcos(?dtf)j
Aestsin(?dtf) XA(t) Aest
This property is useful in calculating system
damping on line and potentially in calculating
PSS signals and signals that might be used to
separate systems that will break apart in
uncontrolled separation.
16
For example
A 0.270 Hz decaying sinusoid, damping factor 0.1
The HT of this signal
The magnitude of the analytic function plotted
on a log scale
17
For example
  • Observations
  • The slope of the log of the XA function is the
    value of s, namely the negative of the damping
    factor, 0.1 in this case
  • The plot is obtained numerically, and only the
    near end values of the plot lie off the line y
    mxb. This is due to end effects of the DFT
    calculation of the HT from a finite sample.
  • Since the HT is in the time domain, if the
    damping changes at time to, the slope of the log
    plot will simply change at time to.
  • Since the DFT is used, as measured data become
    available, the oldest datum is simply dropped out
    of the DFT calculation, and the new datum is
    brought in in the fashion of a sliding window.

The magnitude of the analytic function plotted
on a log scale
18
The phase of the analytic function
XA(t) Aestcos(?dtf)j Aestsin(?dtf) Arg(XA(t)
) ?dtf
This is ?d
19
A synthetic example
A synthetic example corrupted by noise (S5
with SNR 2, S6 with SNR 5). The base
signal S5 is augmented with a second mode at 0.6
Hz, unity amplitude, time constant 8 s in S6.
S5
Prony analysis
Signal Component (Hz) Frequency Identified (Hz) Frequency Identified (Hz) Attenuation factor identified (s) Attenuation factor identified (s)
Signal Component (Hz) 0 50s 0 10s 0 50s 0 10s
S5 0.27 0.271 0.272 9.3 7.2
S6 0.27 0.60 0.272 0.604 N/A 0.566 5.1 6.4 N/A 2.4
Hilbert analysis
Signal Component (Hz) Frequency Identified (Hz) Frequency Identified (Hz) Attenuation factor identified (s) Attenuation factor identified (s)
Signal Component (Hz) 0 50s 0 10s 0 50s 0 10s
S5 0.27 N/A N/A N/A N/A
S6 0.27 0.60 N/A N/A 0.272 0.602 N/A N/A 8 10
20
Actual signal taken in a power system after a
large disturbance
  • Successively zoomed traces
  • Prony sees potentially spurious modes the
    number is selected by the user
  • Hilbert beats Prony in computational speed
  • Hilbert can identify changes in modes as the
    event unfolds
  • Prony assumes stationarity in the signal
  • Prony has been programmed in commercially
    available packages readily used
  • Accuracy is similar between Prony and Hilbert

M1 A measured signal
Method Component Frequency (Hz) Damping ratio Comment on amplitude
Prony 1 2 3 4 0.23 0.30 0.49 0.77 0.001 0.72 0.012 0.023 Dominant Minor Minor Negligible
Hilbert 1 0.23 -0.003 Sole
21
Bases of assessing the tools used for power
system signal processing
  • Multiple modes and modes that are near each other
  • Noise in the measurements
  • Missing measurements
  • Finite sample of the time domain signal (finite
    time window)
  • Three phase issues
  • Speed of the identification can it be done in
    real time?
  • Suitability for control action
  • Accuracy of the identification

22
Execution speed
A second measured signal M2 These are tie line
flows Zoomed traces
  • 100 identifications
  • On-line capability
  • Hilbert generally beats Prony in speed
  • Accuracy in synthetic signals appears to be about
    the same
  • Does not include preprocessing

23
Time domain windowing
Windowing may be viewed as multiplication by a
rectangular pulse p(t). Thus the signal measured
is not x(t), but p(t)x(t)
Time domain windowing will impact both Prony and
Hilbert analysis. The impact on Prony can not be
corrected, but there is potential for correction
in the Hilbert domain.
1
0
24
Bedrosians theorem
Sample length T
Period of oscillation To
  • The signal x(t) is known only in a finite time
    window 0,T
  • The Hilbert transform is x(t)p(t) where p(t) is a
    rectangular pulse that captures that window
  • The Hilbert transform is Hx(t)p(t)
    p(t)Hx(t) p(t)X(t) for pulse widths that are
    significant relative to the period of oscillation
    of x(t) , To ltlt T
  • This approximation is Bedrosians theorem and it
    is a consequence of a narrow band model
  • Under the narrow band model, X(t) changes from
    cosine forms to sine forms, and the angle of the
    analytic function of x(t) is calculated
    accurately from the arg(XA(t))

25
Bedrosians theorem
It would be nice to reduce T as much as possible.
This can be done via several routes
Sample length T
Period of oscillation To
Reduce the BW of the signal
Preprocess data
Modulate the signal with a sweeping frequency
Capture data
Noise filters
Separate even and odd parts of x(t) Work with
moving time window and process only changes in
X(t) Combine with wavelet analysis
Remove the assumptions of Bedrosian's theorem
Splines to envelope the signal
?
Remove high frequency signals and process
separately
26
Hilbert Huang method
  • Effectively reduces the BW of x(t) and allows
    high speed processing of individual component
    bands of frequencies
  • Programmed in a commercial prototype, and proven
    in a range of applications
  • Although not based on Bedrosians theorem, the HH
    method breaks the signal x(t) into component band
    limited signals, and processes those separately.
    The HHT method uses splines and time domain
    sifting. These are similar to demodulation.
    The preprocessing is in the time domain.

Preprocess data
Modulate the signal with a sweeping frequency
Capture data
Noise filters
Use peaks to demodulate the signal
Splines to envelope the signal
Remove high frequency signals and process
separately
27
Hilbert Huang method
SPLINES
  • The basic idea is to develop a series of splines
    that span time intervals 1, 2, , k, such that
    the signal is stationary within the spline
    horizon.
  • Then subtract a projected modal function within
    each spline horizon, m1(t) x(t) h1(t), m11(t)
    h1(t)-h11(t),
  • Stop subtracting estimated modal functions when
    the Cauchy convergence test is satisfied, and
    repeat over all splines

C is sufficiently small as set by the user. This
is effectively a nonlinear low pass filter
28
The challenges
Fully exploit Bedrosians theorem
Bedrosians theorem seemingly allows the use of
shortened time windows of data if the product
p(t)x(t) accounts for the rectangular pulse p(t)
. There have been published ways to handle
products such as this but no one has fully
exploited the results. It is possible that much
shorter clips of data would be useable in
obtaining intrinsic power system modes. A side
benefit if Bedrosians theorem is applied to
band limited signals, the convolution property
results but it is in the time domain.
Combine the Prony and Hilbert methods
The Prony method has been programmed,
commercialized and widely used for many years
and there are many proponents of the method. The
Hilbert method may be viewed as a competitor by
some. But there are real possibilities to use
the time specific properties of Hilbert to size
the sample window for Prony, or to obtain
accurate results for nearly collocated modes, or
to simply obtain a second estimate which may be a
sanity check.
29
The challenges
Apply Titchmarsh's theorem
Titchmarshs theorem if f(t) is square
integrable over the real axis, then any one of
the following implies the other two 1. The FT,
F, is 0 for negative time 2. In the FT, replacing
? by xjy, results in a function that is analytic
in the complex plane and its integral is bounded.
3. The real and imaginary parts of F(xjy) are
the HTs of each other This theorem may allow one
to calculate the HT very rapidly by construction
of the analytic function of f(t). Also, there
are some consequences of autofiltering of f(t)
working in the Hilbert domain.
Solve the Riemann Hilbert problem for this
application
Form an analytic function from the even and odd
parts of a signal fe(t) and fo(t) namely
M(t)fejfo. Then consider two additional
functions a(t) and b(t) such that afe-bfo c.
The question is to find a and b such that the
even part of M(z) where z replaces t and z is a
complex number is the HT of c(t). This may
allow the selection of functions a and b that are
band limited and this will allow rapid
calculation of the HT of f. And this may allow
extraction of the component modes of f(t).
30
The challenges
Perform an error analysis for the HT and HT to
quantify the accuracy of the methods
All practical uses of the HT actually use the
discrete HT. The DHT is obtained from the DFT.
For an n-point implementation, there is a known
error introduced in the DHT calculation. This
implies that some kind of error correction may be
possible. The DFT calculation error for one type
of signal is shown in red.
Apply the method for large scale, high profile
applications
The HT method has been applied in laboratory
controlled circumstances. The need to is to
apply the idea in large systems with many
intrinsic modes. And implement the calculation
alongside a Prony calculation. And also to make
the HT calculation an option in commercial
software.
31
Some additional potential applications of the HT
Hilbert Transformers This is a pair of digital
filters that generates outputs u(t) and v(t)
given an input x(t) where u and v are in
quadrature that is, their joint integral is
zero.
All pass H1(jf)
u(t)
x(t)
v(t)
All pass H2(jf)
Application In real time generate a voltage that
is the q-axis component of a three phase signal,
and use a power electronic amplifier to generate
a signal xq(t) which is injected in series with
the supply for power conditioning.
Electronic waveform generator
x(t)
Hilbert Phase Modulation Phase modulation occurs
inadvertently in bus voltages, at low
frequencies, due to power swings. The HT of a
phase modulated signal is of the form
Jn(ß)ej2pnat , Jn is a Bessel function, 2pna is
the frequency of the phase modulation. This HT
can be calculated easily in real time, and it may
be possible to inject a signal into the
transmission system to cancel interarea
oscillations.
HT
Bessel function look up table
Power system stabilizer
Application a power system stabilizer
32
Contributors to the Hilbert method of signal
analysis
Edward Charles Titchmarsh 1899 1963 U. K.
Georg Friedrich Bernhard Riemann 1826
1866 Germany
?? Norden E. Huang 1942 - Taiwan
Edward Bedrosian 1922 - U. S. A.
33
More information
  • N. E. Huang, Z. She, S. R. Long, M. C. Wu, S. S.
    Shih, Q. Zheng, N.-C. Yen, C. C. Tung, H. H. Liu,
    The empirical mode decomposition and the Hilbert
    spectrum for nonlinear and non-stationary time
    series analysis, Proc. Royal Society of London,
    vol. 454, pp. 903-995, 1998.
  • S. L. Hahn, Hilbert Transforms in Signal
    Processing, Boston, Artech House, 1996.
  • J. Hauer, D. Trudnowski, G. Rogers, B.
    Mittelstadt, W. Litzenberger , J. Johnson,
    Keeping an eye on power system dynamics, IEEE
    Computer Applications in Power, vol. 10, No. 4,
    pp. 50-54, Oct. 1997.
  • A. R. Messina, V. Vittal, D. Ruiz-Vega, G.
    Enríquez-Harper, Interpretation and
    visualization of wide-area PMU measurements using
    Hilbert analysis, IEEE Transactions on Power
    Systems, vol. 21, No. 4, pp. 1763-1771, Nov.
    2006.
  • Timothy Browne, V. Vittal, G. T. Heydt, Arturo R.
    Messina, A real time application of Hilbert
    transform techniques in identifying inter-area
    oscillations, Chapter 4, Interarea Oscillations
    in Power Systems, Springer, New York NY, 2009,
    pp. 101 125

34
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