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The Hilbert Transform: Applications in the Analysis of Power Engineering

Dynamics

G. T. HeydtArizona State University

October, 2009

Presentation at the Missouri University of Science and Technology

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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

PSERCObjectives

• 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

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PSERCIl existe de nombreuses façons d'afficher une

image

What is an ‘image’

5

•A way to see something•A view not easily interpreted otherwise

•Trans = across Form = manifestation

TRANSFORMATIONA mapping from

one space to another

transform

PSERCIl existe de nombreuses façons d'afficher une

image

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TRANSFORMATIONA mapping from

one space to another

• The concept is to make calculations easier in the transformed domain

• And not to waste too much time in transforming and untransforming

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Issues in power signal identification

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

Transforms are often useful for these applications

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Why use transformations?

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 easierAnd to conform with widely used notation

Fourier transformLaplace transformHartley transform

Fourier transformLaplace transformHartley transform

Discrete Hartley and Fourier transforms

Walsh transform

Symmetrical components,

Clarke’s components

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David Hilbert

1862 – 1943born in Königsberg, East

Prussiaalgebraic formsalgebraic number theoryfoundations of geometryDirichlet's principlecalculus of variationsintegral equationstheoretical physics and

dynamicsfoundations of

mathematicsthe Hilbert transform

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)t(x*t

1)t(X)]t(x[H

The Hilbert transform

Some points of interestThe transformed variable is still tThe convolution integral is best performed by taking the FT of both sides – and use the convolution property of the FTRecall 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

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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

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)t(x*t

1)t(X)]t(x[H

The Hilbert transform

These are FT

transform pairs

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)t(x*t

1)t(X)]t(x[H

The Hilbert transform

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).

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Some rather interesting Hilbert transforms

)tdcos(tAe )tdsin(tAe

x(t) X(t)

Special interest in dynamic

studies of all kinds of linear

systems

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Some rather interesting properties of the Hilbert transform

Linearity

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)]

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

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The analytic function of the decaying sinusoid

XA(t) = x(t) + j H[x(t)]

)tdcos(tAe )tdsin(tAe

Therefore XA(t) = Aeσtcos(ωdt+φ)+j Aeσtsin(ωdt+φ)|XA(t)|= Aeσt

The analytic function

HT pair

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.

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For example

A 0.270 Hz decaying sinusoid, damping

factor 0.1The HT of this

signal

The magnitude of the analytic function – plotted on a log scale

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For example

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

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

Arg(XA(t)) = ωdt+φ

The phase of the analytic function

This is ωd

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Signal Component (Hz)

Frequency Identified (Hz)

Attenuation factor identified (s)

0 – 50s 0 – 10s 0 – 50s 0 – 10s

S5 0.27 0.271 0.272 9.3 7.2S6 0.27

0.600.2720.604

N/A0.566

5.16.4

N/A2.4

Signal Component (Hz)

Frequency Identified (Hz)

Attenuation factor identified (s)

0 – 50s 0 – 10s 0 – 50s 0 – 10s

S5 0.27 N/A N/A N/A N/AS6 0.27

0.60N/AN/A

0.2720.602

N/AN/A

810

noise ) t27.0π 2(ef(t) t/10 sin

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.

Prony analysis

Hilbert analysis

S5

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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 HilbertMethod Component

Frequency (Hz)

Damping ratio

Comment on

amplitudeProny 1

234

0.230.300.490.77

0.0010.72

0.0120.023

DominantMinorMinor

Negligible

Hilbert 1 0.23 -0.003 Sole

M1 A measured signal

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•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

Bases of assessing the tools used for power system signal processing

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•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

Execution speed

A second measured signal: M2These are tie line flowsZoomed traces

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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.

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

1

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0

1

•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))

Bedrosian’s theorem

Period of oscillation To

Sample length T

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Separate even and odd parts of x(t)Work with moving time window and process only changes in X(t)Combine with wavelet analysis 25

0

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

Bedrosian’s theorem

Period of oscillation To

Sample length T

Capture data

Preprocess data

Reduce the BW of the

signal

Remove the assumption

s of Bedrosian's

theorem

Noise filters

Splines to envelope the

signal

Modulate the signal with a

sweeping frequency

Remove high frequency signals

and process separately

?

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Hilbert Huang method

Capture data

Preprocess data

Noise filters

Splines to envelope the

signal

Modulate the signal with a

sweeping frequency

Remove high frequency signals

and process separately

•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.

Use peaks to demodulate the signal

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Hilbert Huang method

•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

SPLINES

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The ‘challenges’

Fully exploit Bedrosian’s theorem

Combine the Prony and Hilbert methods

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.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.

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The ‘challenges’

Apply Titchmarsh's theorem

Solve the Riemann – Hilbert problem for this application

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 time2. 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.

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).

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The ‘challenges’

Perform an error analysis for the HT and HT to quantify the accuracy of the methods

Apply the method for large scale, high profile applications

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.

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.

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Some additional potential applications of the HT

x(t)All pass H1(jf)

All pass H2(jf)

Hilbert Phase ModulationPhase 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(β)ej2πnat , Jn is a Bessel function, 2πna 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.

u(t)

v(t)

ApplicationIn 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 TransformersThis 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.

HT

Bessel function look up table

x(t)

Electronic waveform generator

Power system stabilizer

Application: a power system stabilizer

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Edward Charles Titchmarsh1899 – 1963

U. K.

Georg Friedrich Bernhard Riemann

1826 – 1866Germany

黃鍔Norden E. Huang

1942 - …Taiwan

Contributors to the Hilbert method of signal analysis

Edward Bedrosian1922 - …U. S. A.

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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

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Questions?

Comments?