1. Oscillation Detection and Modal Analysis of Ambient Data
Bernie Lesieutre
University of Wisconsin - Madison
Dept. of Electrical & Computer Engineering
WECC, SLC, March 4, 2015

2. Oscillation Detection
We 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.

3. Signal Detection with Matched Fileter
detection filter
s(t) or
r(t)
v(t)
ρ(t) > γ s(t) detected
ρ(t) < γ s(t) not detected
h(t)
ρ(t)
decision
(ROC, etc)
signal
detection signal
noise
8x10-3 4
detection signal
signal
noise and signal

4. Oscillation Detection
Candidate “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 , 0.25, 0.40, 0.67, 1.25, and Hz.

5. Frequency Detection Stripchart
2.00
1.25
0.67
0.40
0.25
0.10

6. Oscillation Detection Stripchart
~0.6 Hz
~8 peak-peak
2.00
1.25
0.67
0.40
0.25
0.10

7. Oscillation Detection Stripchart
~0.25 Hz
~5 peak-peak
2.00
1.25
0.67
0.40
0.25
0.10

8. Next Step
Next, design characteristics of detection filters that allow correlating detection signals to better distinguish oscillation frequency and amplitude. f
filter characteristics

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

10. 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.

11. Model Fit, example
However, many typical approaches use a three-stage process:
Use correlations in data to construct a linear system model.
Calculate natural modes of model. Roots of
Calculate corresponding coefficients to match data.
Advantage: Each step involves a FAST linear calculation.

12. Curve Fitting
Fit data to (un)damped sinusoids
Mode Shapes

13. Nonlinear Method Variable Projection Method
Optimization variables (damping & frequencies)
Basis functions (sinusoids, exponentials, polynomial (trend))
Variable Projection Method
“The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate,” Golub and Pereyra (1973)
Gradient:

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

15. Disturbance: Angle Difference Data
Ringdown Analysis (varpro)
% damping
% damping
% damping
10 seconds of ringdown data, scaled and shift.
Varpro fit to the data.

16. Ambient Data
Use five minutes of data prior to disturbance to estimate modes:
Is there any information there?
Estimate using Varpro fit to sample autocovariances

17. Ambient Data
Five minutes of data (scaled and shifted)
FFT of data

18. Autocovariance Fit Ringdown Analysis (varpro) 0.32 Hz @ 9% damping
Promising start …