Real Time Phasor Data Processing using
Complex Number Analysis Methods
Raymond de Callafon, Charles H. Wells
University of California, San Diego & OSIsoft
CIGRE Grid of the Future Symposium, Oct. 11-13, 2015, Chicago, IL
email: [email protected], [email protected]
UCSD Phasor Measurement System
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Multiple PMUs
currently on UCSD
campus
Installation of 20
additional microPMUs
Lot of data being
generated
Objectives:
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Automatically detect
and mark events
When event occurs,
model dynamics
CIGRE GoTF Symposium 2015, Callafon & Wells
Research and new OSIsoft lab at UCSD
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Current research on phasor data analysis at System
Identification and Control Laboratory (SICL) at UCSD
Current collaborations with
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New development: new data acquisition laboratory at
San Diego Supercomputer Center
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Center for Energy Research (CER)
OSIsoft
SDG&E
Sponsored by OSIsoft
Streaming of PMU data
(data quality!)
Real-time applications in signal
processing and data mining
CIGRE GoTF Symposium 2015, Callafon & Wells
Outline
Outline of this Talk
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Phasor Data
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Application of complex Discrete Fourier Transform (DFT)
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(Voltage, Current or Power) data analysis
Use of complex phasor data (real/imag part)
Well known facts on DFT
Results for complex DFT
Application of moving complex DFT to actual phasor
data for event detection
CIGRE GoTF Symposium 2015, Callafon & Wells
Phasor Data Analysis
PMU data (Frequency/Voltage/Power) to observe events
We are used to using
Frequency, Voltage,
(unwrapped) Angle to
observe events.
Example: May 12 Oakland PMU
from 10:30 – 10:45am data
Are there alternatives
signals for event
detection?
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CIGRE GoTF Symposium 2015, Callafon & Wells
Complex Phasor Data Analysis
Obvious choice would be
the time varying phasor
(real/imag data) directly
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imag part 𝑥𝑖 𝑘
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Top figure: real/imag
No need to “unwrap”
phase (bottom figure)
𝑥 𝑘
angle
real part 𝑥𝑟 𝑘
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CIGRE GoTF Symposium 2015, Callafon & Wells
Voltage Phasor Data
Complex Phasor Data Analysis
Obvious choice would be
the time varying phasor
(real/imag data) directly
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Top figure: real/imag
No need to “unwrap”
phase (bottom figure)
Possible to process
“filtered rate of change”
(FRoC) of phasor via 2D
analysis
CIGRE GoTF Symposium 2015, Callafon & Wells
Complex Phasor Data Analysis
Obvious choice would be
the time varying phasor
(real/imag data) directly




8
Top figure: real/imag
No need to “unwrap”
phase (bottom figure)
Possible to process
“filtered rate of change”
(FRoC) of phasor via 2D
analysis
Allows 3D visualization of phasor data and events
CIGRE GoTF Symposium 2015, Callafon & Wells
DFT of Complex Phasor Data Analysis
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Consider phasor 𝑥 𝑘 = 𝑥𝑟 𝑘 + 𝑗 ∙ 𝑥𝑖 (𝑘) where 𝑥𝑟 𝑘 is real part
and 𝑥𝑖 (𝑘) is imaginary part
Let 𝑋𝑟 (𝑛) and 𝑋𝑖 (𝑛) be the N point DFT of the real-valued signals
𝑥𝑟 𝑘 and 𝑥𝑖 𝑘 respectively.
Well known fact(s):
IF 𝑋𝑟 𝑛 = 𝑎 + 𝑗𝑏 AND 𝑋𝑖 𝑛 = 𝑐 + 𝑗𝑑 THEN we know
 𝑋𝑟 𝑛 + 𝑁/2 = 𝑎 − 𝑗𝑏
 𝑋𝑖 𝑛 + 𝑁/2 = 𝑐 − 𝑗𝑑
Interpretation:
First and last N/2 points 𝑋𝑟 (𝑛) and 𝑋𝑖 (𝑛) are complex conjugate
Note: choose 2log(n) = integer to allow Fast Fourier Transform (FFT)
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CIGRE GoTF Symposium 2015, Callafon & Wells
DFT of Complex Phasor Data Analysis
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Again, consider phasor 𝑥 𝑘 = 𝑥𝑟 𝑘 + 𝑗 ∙ 𝑥𝑖 (𝑘)
Let 𝑋(𝑛) be a N point “complex DFT” of complex phasor
𝑁
𝑥 𝑘 𝑒 −𝑗2𝜋𝑘𝑛/𝑁
𝑋 𝑛 =
𝑘=1
Let 𝑋𝑟 (𝑛) and 𝑋𝑖 (𝑛) be the N point DFT of 𝑥𝑟 𝑘 and 𝑥𝑖 𝑘
Interesting result:
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𝑋𝑟 𝑛 = 𝑎 + 𝑗𝑏 𝑋𝑟 𝑛 + 𝑁/2 = 𝑎 − 𝑗𝑏
IF
(previous well-known fact)
𝑋𝑖 𝑛 = 𝑐 + 𝑗𝑑 𝑋𝑖 𝑛 + 𝑁/2 = 𝑐 − 𝑗𝑑
THEN complex DFT 𝑋(𝑛) of 𝑥(𝑘) satisfies
 𝑋 𝑛 = 𝑋𝑟 𝑛 + 𝑗 ∙ 𝑋𝑖 𝑛 = 𝑎 − 𝑑 + 𝑗(𝑐 + 𝑏)
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𝑋 𝑛 + 𝑁/2 = 𝑋𝑟 𝑛 + 𝑁/2 + 𝑗 ∙ 𝑋𝑖 𝑛 +
CIGRE GoTF Symposium 2015, Callafon & Wells
𝑁
2
= 𝑎 + 𝑑 + 𝑗(𝑐 − 𝑏)
DFT of Complex Phasor Data Analysis
What does this mean?
 𝑋 𝑛 = 𝑋𝑟 𝑛 + 𝑗 ∙ 𝑋𝑖 𝑛 = 𝑎 − 𝑑 + 𝑗(𝑐 + 𝑏)
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𝑋 𝑛 + 𝑁/2 = 𝑋𝑟 𝑛 + 𝑁/2 + 𝑗 ∙ 𝑋𝑖 𝑛 +
𝑁
2
= 𝑎 + 𝑑 + 𝑗(𝑐 − 𝑏)
Interpretation and combinatorics:
 The DFT 𝑋(𝑛) will not have the same complex conjugate property
as the individual DFTs 𝑋𝑟 (𝑛) and 𝑋𝑖 (𝑛), e.g. 𝑋 𝑛 + 𝑁/2 ≠ 𝑋 𝑛 ∗ .
 However, the DFT 𝑋(𝑛) will be a combination of the real and
imaginary parts of the individual DFTs 𝑋𝑟 (𝑛) and 𝑋𝑖 (𝑛), e.g.
𝑅𝑒{𝑋 𝑛 } equals 𝑅𝑒 𝑋𝑟 𝑛 − 𝐼𝑚{𝑋𝑖 𝑛 }.
 Moreover, the individual DFTs 𝑋𝑟 (𝑛) and 𝑋𝑖 (𝑛) can be
reconstructed from the DFT 𝑋(𝑛) and 𝑋(𝑛 + 𝑁/2), e.g.
𝑅𝑒 𝑋𝑅 𝑛 = 1/2𝑅𝑒 𝑋 𝑛 + 1/2𝑅𝑒 𝑋 𝑛 + 𝑁/2 .
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CIGRE GoTF Symposium 2015, Callafon & Wells
DFT of Complex Phasor Data Analysis
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Moving N point
“complex DFT”
of (rate of change) of
complex phasor 𝑥 𝑘
−𝑗2𝜋𝑘𝑛/𝑁
𝑋 𝑛 = 𝑁
𝑘=1 𝑥 𝑘 𝑒
Inspect |𝑋 𝑛 | as usual
Since you evaluate both
real/imag, you will see
disturbance effects in
both angle/frequency
and amplitude
CIGRE GoTF Symposium 2015, Callafon & Wells
Voltage Phasor Data
Application to Phasor Data
Conclusion:
it is computationally
advantageous to compute
complex FFTs on phasor
data (real/imag) directly
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Top figure: filtered
rate of change of
real/imag of phasor
No need to “unwrap”
phase!
Bottom figure: moving
FFT analysis of complex
phasor
CIGRE GoTF Symposium 2015, Callafon & Wells
Moving DFT
Summary
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Detection of Events via Filtered Rate of Change (FRoC)
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Auto Regressive Moving Average (ARMA) filter of ambient data
Rate-of-Change filter to create FRoC signal to detect change
Can be applied to Frequency/Voltage/unwrapped Angle
Can also be applied directly to phasor data (Real/Imag)
instead of (Amplitude/Angle)
Ring down analysis via Realization Algorithm
CIGRE GoTF Symposium 2015, Callafon & Wells