Oscillation Detection and Modal Analysis of Ambient Data

Bernie LesieutreUniversity of Wisconsin - MadisonDept. of Electrical & Computer Engineering

WECC, SLC, March 4, 2015

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Oscillation DetectionWe want to detect oscillations quickly.We could use FFTs for detection … but then we are beholden to the lowest frequency of interest, which slows detection of higher frequencies.Furthermore, if we know a priori the frequencies of interest, we can focus on detecting those. This leads to matched filters.

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Signal Detection with Matched Fileter

s(t) or 0

r(t)

v(t)

ρ(t) > γ s(t) detectedρ(t) < γ s(t) not detected 

h(t) ρ(t)

signalnoise

detection filter

detection signal

decision(ROC, etc)

signal

8x10-3

noise and signal

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detection signal

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Oscillation DetectionCandidate “matched” Filters for detecting 1.25 Hz. It matches two cycles of a sinusoidal waveform… and with hamming window

Initial Approach: form a set of filters centered on certain frequencies. For illustration here, use 0.10, 0.25, 0.40, 0.67, 1.25, and 2.00 Hz.

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Frequency Detection Stripchart

2.001.250.67 0.400.250.10

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Oscillation Detection Stripchart

~0.6 Hz~8 peak-peak2.00

1.250.67 0.400.250.10

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Oscillation Detection Stripchart

2.001.250.67 0.400.250.10

~0.25 Hz~5 peak-peak

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Next Step

•Next, design characteristics of detection filters that allow correlating detection signals to better distinguish oscillation frequency and amplitude.

ffilter characteristics

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Ambient Data Analysis

“Modal” Analysis of Data SignalSomething akin to Fourier Analysis except using damped sinusoids to represent signal.

“Modal” Analysis of Ambient Data SignalSomething akin to Fourier Analysis except using damped sinusoids to represent the autocovariance signal of the ambient data.

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Modal Analysis Approaches

•Model Fitting: explicitly or implicitly construct (linear) model. Fit data to basis functions based on the natural modes of the model.

•Curve Fitting: determine parameters of parameterized basis functions and fit.

- FFT, polynomials, varpro

FAST! Straightforward Linear Calculations!

Generally a nonlinear optimization for exponential basis functions.

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Model Fit, example

However, many typical approaches use a three-stage process:1. Use correlations in data to construct a linear system

model.2. Calculate natural modes of model. Roots of

3. Calculate corresponding coefficients to match data.

Advantage: Each step involves a FAST linear calculation.

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Curve Fitting

Mode Shapes

•Fit data to (un)damped sinusoids

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Nonlinear Method

▫ Variable Projection Method− “The Differentiation of Pseudo-Inverses and Nonlinear Least

Squares Problems Whose Variables Separate,” Golub and Pereyra (1973)

Optimization variables (damping & frequencies)Optimization variables (damping & frequencies)

Basis functions (sinusoids, exponentials, polynomial (trend))

Gradient:

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Modal Analysis of Ambient DataWe want to detect the possibility of poorly damped oscillations before an event triggers them.

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Disturbance: Angle Difference Data

Ringdown Analysis (varpro)

0.32 Hz @ 9% damping0.67 Hz @ 13 % damping0.87 Hz @ 10 % damping

10 seconds of ringdown data, scaled and shift.Varpro fit to the data.

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Ambient Data

Use five minutes of data prior to disturbance to estimate modes:

1. Is there any information there?2. Estimate using Varpro fit to sample autocovariances

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Ambient Data

Five minutes of data (scaled and shifted) FFT of data

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Autocovariance Fit

Ringdown Analysis (varpro)

0.32 Hz @ 9% damping0.75 Hz @ 0 % damping0.87 Hz @ 10 % damping Promising start …