Presentation at the Missouri University of Science and
Technology

PSERC

The Hilbert Transform: Applications in
the Analy...
Я

2фӘ-Я/‫ﮎ‬Ǽ4

It is obvious that …

PSERC

Ǽ5
Я*

μφ21 = 2π±σg+Q=buuY(t) >> 12.1028746*x(z)

Ξ2-3Ω8>1

ξΩ

Except
πΔΨ = ...
Outline

PSERC

• Why look to transform theory for any help in
power system dynamic analysis?
• The Hilbert transform
• So...
Objectives

PSERC

• To introduce the Hilbert transform in a comprehensible
way
• To discuss applications in power enginee...
Il existe de nombreuses façons d'afficher une
image

PSERC

transform
What is an ‘image’
• A way to see something
• A view...
Il existe de nombreuses façons d'afficher une
image

PSERC

• The concept is to make calculations easier in the
transforme...
Issues in power signal identification

PSERC

Transforms are often useful
for these applications

POWER
SYSTEM

Measuremen...
Why use transformations?
Fourier transform
Laplace transform
Hartley transform

To convert a differential equation to an
a...
David Hilbert

PSERC

1862 – 1943
born in Königsberg, East Prussia
algebraic forms
algebraic number theory
foundations of ...
The Hilbert transform

PSERC

1
H [ x( t )] = X ( t ) =
* x( t )
πt
Some points of interest
The transformed variable is st...
The Hilbert transform

PSERC

1
H [ x( t )] = X ( t ) =
* x( t )
πt
The FT of the 1/πt term is –jsgn(ω)
This can be verifi...
The Hilbert transform

PSERC

1
H [ x( t )] = X ( t ) =
* x( t )
πt
Therefore, one way to obtain the HT is to MULTIPLY the...
Some rather interesting Hilbert
transforms
x(t)

Aeσt cos( ω d t + ϕ )

PSERC

X(t)

Special interest in
dynamic studies o...
Some rather interesting properties of the
Hilbert transform
Linearity

H(ax(t))=aX(t)

PSERC

H(x(t)+y(t))=X(t)+Y(t)

Doub...
The analytic function of the decaying
sinusoid
Aeσt cos( ω d t + ϕ )

σt
Ae sin( ωd t + ϕ )

The analytic function

PSERC
...
For example

A 0.270 Hz decaying
sinusoid, damping factor 0.1

PSERC

The HT of this signal

The magnitude of the analytic...
For example

PSERC

Observations
•The slope of the log of the |XA| function is the value of σ, namely the negative of the
...
The phase of the analytic function

PSERC

XA(t) = Aeσtcos(ωdt+φ)+j Aeσtsin(ωdt+φ)
Arg(XA(t)) = ωdt+φ

This is ωd

18
A synthetic example
A synthetic example – corrupted
by noise (‘S5’ with SNR = 2, ‘S6’
with SNR = 5). The base signal S5
is...
Actual signal taken in a power system
after a large disturbance
M1 A measured signal

PSERC

•Successively zoomed traces
•...
Bases of assessing the tools used for
power system signal processing

PSERC

•Multiple modes and modes that are near each ...
Execution speed

PSERC

A second measured signal: M2
These are tie line flows
Zoomed traces

•100 identifications
•On-line...
Time domain windowing

Time domain windowing will impact both Prony
and Hilbert analysis. The impact on Prony can
not be c...
Bedrosian’s theorem
1

PSERC

Sample length T

0
Period of oscillation To

•The signal x(t) is known only in a finite time...
Bedrosian’s theorem
1

It would be nice to
reduce T as much
as possible. This
can be done via
several routes

Sample lengt...
Hilbert Huang method

PSERC

•Effectively reduces the BW of x(t) and allows high speed processing of individual
component ...
Hilbert Huang method

PSERC

SPLINES

•The basic idea is to develop a series of splines that span time intervals 1, 2, …, ...
The ‘challenges’

PSERC

Fully exploit Bedrosian’s theorem
Bedrosian’s theorem seemingly allows the use of shortened time ...
The ‘challenges’

PSERC

Apply Titchmarsh's theorem
Titchmarsh’s theorem: if f(t) is square integrable over the real axis,...
The ‘challenges’

PSERC

Perform an error analysis for the HT and HT to quantify the
accuracy of the methods
All practical...
Some additional potential applications of the HT
Hilbert Transformers
This is a pair of digital filters that generates out...
Contributors to the Hilbert method of signal
analysis
Georg Friedrich
Bernhard Riemann
1826 – 1866
Germany

黃鍔
Norden E. H...
More information
PSERC

•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...
PSERC

Questions?

Comments?

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Seminar 20091023 heydt_presentation

  1. 1. Presentation at the Missouri University of Science and Technology PSERC The Hilbert Transform: Applications in the Analysis of Power Engineering Dynamics G. T. Heydt Arizona State University October, 2009
  2. 2. Я 2фӘ-Я/‫ﮎ‬Ǽ4 It is obvious that … PSERC Ǽ5 Я* μφ21 = 2π±σg+Q=buuY(t) >> 12.1028746*x(z) Ξ2-3Ω8>1 ξΩ Except πΔΨ = Ω+Ξ/0.24r ∩∏5≠≈Ǿ*Ǽ *ф/Ө-1 1.1111+Єζ/ξΩ√ќ when n is q34-atan(acos(z))Λcosh(q)/Ж2=0.2957352957Юq odd or n = 57Юq 8ф21/Ө-1 = ‫ﻅ‬ 3 ∫∫∫ 1279 g k − ∂r + π  1 0 0   − 1 0 0 0 ∆Ξ  − 1 0.111  12λφ = k = 22 0 Γ  * * − Χ 0 Γ − 1.1t / 2 +  Χ 0 Γ * 202 − 1.1t Χ ∑ i ∇ℑϖ ΠΘ Ξ + 1.1 ΠΘ Ξ + 1.1  0 ΠΘ Ξ   0  0        i =ℵ ∑ PFq q =104769 b⊆Ξ φ cosh( a tan(sinh( t * z ))) ∑ λi = i =ℵ Note  ∫∫∫ ∫∫∫∫∫∫ α g k − ∂r + π  1 0  k = 22 y 0 ∇ℑϖ  ∑ ∆Fq 0 Θ  q =104769    22.5  2  Χ − 2* Γ  φ −∃ Ξ 22 ∑ 12k   k =qw+5  1 ℜ ∏ −16.7 qw− 2 * q  y a ⊗b ∆r = Ξ 8 qq( φµπξωγ ) + 14v * * ô/44 - 1 c⊕d 5vô/2 - 1 ∫∫∫ ∫∫∫∫∫∫ α g k − ∂r + π k = 22 ∇ℑϖ ∑ ∆Fq q =104769 VERY IMPORTANT FOR Z = 2₣‫ שׂ‬Я 2 3 Λ ΣΠ q⊥r ℑ 6 7 4 Ψ s2 8 2
  3. 3. Outline PSERC • 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. 4. Objectives PSERC • 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 4
  5. 5. Il existe de nombreuses façons d'afficher une image PSERC 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 5
  6. 6. Il existe de nombreuses façons d'afficher une image PSERC • 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 6
  7. 7. Issues in power signal identification PSERC Transforms are often useful for these applications POWER SYSTEM Measurements IDENTIFICATION Take corrective control action, alarms, PSS signals The main contenders • Fourier analysis • Prony analysis • Hilbert analysis • Various control theory approaches such as observer design 7
  8. 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 Fourier transform Laplace transform Hartley transform Discrete Hartley and Fourier transforms To get rid of unbalanced three phase quantities To make calculations easier And to conform with widely used notation PSERC Walsh transform Symmetrical components, Clarke’s components 8
  9. 9. David Hilbert PSERC 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 9
  10. 10. The Hilbert transform PSERC 1 H [ x( t )] = X ( t ) = * x( t ) πt 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(ω) This can be verified by the reciprocity theorem: if f(t) and F(jω) are transform pairs, then f(jω) and F(t) are also transform pairs 10
  11. 11. The Hilbert transform PSERC 1 H [ x( t )] = X ( t ) = * x( t ) πt The FT of the 1/πt term is –jsgn(ω) This can be verified by the reciprocity theorem: if f(t) and F(jω) are transform pairs, then f(jω) and F(t) are also transform pairs These are FT transform pairs 11
  12. 12. The Hilbert transform PSERC 1 H [ x( t )] = X ( t ) = * x( t ) πt Therefore, one way to obtain the HT is to MULTIPLY the FT of 1/πt (namely –sgn(ω)) 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 ω. Then take the IFT if you really need X(t). 12
  13. 13. Some rather interesting Hilbert transforms x(t) Aeσt cos( ω d t + ϕ ) PSERC X(t) Special interest in dynamic studies of all kinds of linear systems Aeσt sin( ωd t + ϕ ) 13
  14. 14. Some rather interesting properties of the Hilbert transform Linearity H(ax(t))=aX(t) PSERC H(x(t)+y(t))=X(t)+Y(t) Double application When the HT is applied twice to x(t), the result is –x(t). This is also called anti-involution. Inverse HT H-1 = -H Differentiation H(dx/dt) = d[H(t)]/dt Convolution H(x*y) = X*y = x*Y The analytic function XA(t) = x(t) + j H[x(t)] 14
  15. 15. The analytic function of the decaying sinusoid Aeσt cos( ω d t + ϕ ) σt Ae sin( ωd t + ϕ ) The analytic function PSERC HT pair XA(t) = x(t) + j H[x(t)] Therefore XA(t) = Aeσtcos(ωdt+φ)+j Aeσtsin(ωdt+φ) |XA(t)|= Aeσt 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. 15
  16. 16. For example A 0.270 Hz decaying sinusoid, damping factor 0.1 PSERC The HT of this signal The magnitude of the analytic function – plotted on a log scale 16
  17. 17. For example PSERC Observations •The slope of the log of the |XA| function is the value of σ, 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 = mx+b. 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 17
  18. 18. The phase of the analytic function PSERC XA(t) = Aeσtcos(ωdt+φ)+j Aeσtsin(ωdt+φ) Arg(XA(t)) = ωdt+φ This is ωd 18
  19. 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. PSERC S5 f(t) = e−t/10sin( 2π 0.27 t ) + noise Prony analysis Signal S5 S6 Frequency Identified (Hz) 0 – 50s 0 – 10s Component (Hz) 0.27 0.27 0.60 0.271 0.272 0.604 0.272 N/A 0.566 Attenuation factor identified (s) 0 – 50s 0 – 10s 9.3 5.1 6.4 7.2 N/A 2.4 Hilbert analysis Signal Component (Hz) S5 S6 0.27 0.27 0.60 Frequency Identified (Hz) 0 – 50s 0 – 10s N/A N/A N/A N/A 0.272 0.602 Attenuation factor identified (s) 0 – 50s 0 – 10s N/A N/A N/A N/A 8 1019
  20. 20. Actual signal taken in a power system after a large disturbance M1 A measured signal PSERC •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 (Hz) Damping ratio 1 2 3 4 0.23 0.30 0.49 0.77 0.001 0.72 0.012 0.023 Comment on amplitude Dominant Minor Minor Negligible 1 0.23 -0.003 Sole Method Component Prony Hilbert Frequency 20
  21. 21. Bases of assessing the tools used for power system signal processing PSERC •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 21
  22. 22. Execution speed PSERC 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 22
  23. 23. Time domain windowing 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 PSERC 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) 0 23
  24. 24. Bedrosian’s theorem 1 PSERC Sample length T 0 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 H[x(t)p(t)] ≈ p(t)H[x(t)] = p(t)X(t) for pulse widths that are significant relative to the period of oscillation of x(t) , To << T •This approximation is Bedrosian’s 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)) 24
  25. 25. Bedrosian’s theorem 1 It would be nice to reduce T as much as possible. This can be done via several routes Sample length T 0 Period of oscillation To Preprocess data Capture data ? Remove the assumptions of Bedrosian's theorem PSERC Reduce the BW of the signal Modulate the signal with a sweeping frequency Noise filters Separate even and odd parts of x(t) Splines to envelope Work with moving time window the signal and process only changes in Remove high frequency X(t) signals and process Combine with wavelet analysis separately 25
  26. 26. Hilbert Huang method PSERC •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 Bedrosian’s 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 Capture data Use peaks to demodulate the signal Modulate the signal with a sweeping frequency Noise filters Splines to envelope the signal Remove high frequency signals and process separately 26
  27. 27. Hilbert Huang method PSERC 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 27
  28. 28. The ‘challenges’ PSERC Fully exploit Bedrosian’s theorem Bedrosian’s 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 Bedrosian’s 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. 28
  29. 29. The ‘challenges’ PSERC Apply Titchmarsh's theorem Titchmarsh’s 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 ω by x+jy, results in a function that is analytic in the complex plane and its integral is bounded. 3. The real and imaginary parts of F(x+jy) 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)=fe+jfo. 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). 29
  30. 30. The ‘challenges’ PSERC 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. 30
  31. 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. x(t) All pass H1(jf) All pass H2(jf) PSERC u(t) v(t) 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’. Hilbert Phase Modulation x(t) HT Electronic waveform generator Phase modulation occurs inadvertently in bus voltages, at low frequencies, due to power swings. The HT of a phase modulated signal is Bessel function of the form Jn(β)ej2πnat , Jn is a Bessel function, look up table 2πna is the frequency of the phase modulation. This HT can be calculated easily in real time, Power system stabilizer and it may be possible to inject a signal into the transmission system to cancel interarea Application: a power system stabilizer 31 oscillations.
  32. 32. Contributors to the Hilbert method of signal analysis Georg Friedrich Bernhard Riemann 1826 – 1866 Germany 黃鍔 Norden E. Huang 1942 - … Taiwan PSERC Edward Charles Titchmarsh 1899 – 1963 U. K. Edward Bedrosian 1922 - … U. S. A. 32
  33. 33. More information PSERC •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. 903995, 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 33
  34. 34. PSERC Questions? Comments? 34

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