Phase Identification of Smart Meters Using a
Fourier Series Compression and a Statistical
Clustering Algorithm
Jeremy Chiu, Albert Wong, James Park
Joe Mahony, Michael Ferri, Tim Berson
IEEE EPEC 2022 – December

Overview
• Motivating Fourier series through regression
• Fourier Series on voltage time series
• Phase identification through clustering
• Future work

Lines of Best Fit – Monthly Dog Food Bill
Given the following monthly food bills, estimate the cost to feed a
20lb dog.
• Coby the Pomeranian -- $60, 9lb
• Benji the mutt -- $35, 14lb
• Tofu the Malti-Poo -- $90, 15.5lb
• Berkley the Mini Eskimo -- $70, 18lb
• Noodle the Doodle -- $85, 37lb

Lines of Best Fit – Monthly Dog Food Bill

𝐶 = 𝑎1 𝑊 + 𝑎0

Lines of Best Fit – Energy vs Temperature

𝐸 = 𝑎0 + 𝑎1 𝑇 + 𝑎2 𝑇 2

More generally,

𝑓 𝑡 = 𝑎0 + 𝑎1 𝑡 + 𝑎2 𝑡 2 + 𝑎3 𝑡 3 + …

Lines of Best Fit <– Sines of Best Fit

The (grossly over-simplified) Fourier series of a function f(t) is
2𝜋
2𝜋
2𝜋
𝑓 𝑡 = 𝑎0 + 𝑎1 sin
𝑡 + 𝑎2 sin
2𝑡 + 𝑎3 sin
3𝑡 + ⋯
𝑃
𝑃
𝑃

Fourier Series – Definition and Forms
Consider a periodic function f(t) with period P.

Noisy Brain Example

Maintz, Twan. "Digital and medical image processing." Universiteit Utrecht (2005), Lecture Notes.

Voltage Data Set
• Voltage data set provided by Harris SmartWorks, an American
utility company
• Used Feeder F (26 meters) and Feeder D (55 meters)
• Each meter is one of three (unknown) phases
• Hourly voltage data, so one meter in June has 24*30=720 entries
• Normalized each time series by dividing by its mean

Overview of Fourier Compression

Fourier Spectrum of Time Series

Fourier Time Series Approximation

Phase Identification Through Clustering
• Clustering Method: Ward D, with dimension-reduced Fourier
coefficients and 3 cluster groups
• Kept 12 Fourier coefficients (of the original 720)
• 3 methods to [in]validate clustering results:
• MDS Visualization
• Time Stability
• Same Transformer, Same Cluster

MDS Visualization - Feeder F

Time Stability - Feeder F

Without compression:

With compression:

July

June

A
B
C
Total

A
12

July
B

C

8
1
13

8

5
5

Total
12
8
6
26

June

A
B
C
Total

A
13

B

C

8
13

8

5
5

Total
13
8
5
26

Same Transformer, Same Cluster ?
Almost for Feeder F
Transformer
1
2051
3840
3965
4611
4702
4711
11917
12187
12188
12298
Feeder F

1
1
1
1
1

June
Cluster
2
8

3

1
4
1
1
1
1
5
13

8

5

Total
9
1
1
1
1
4
1
1
1
1
5
26

Transformer
1
2051
3840
3965
4611
4702
4711
11917
12187
12188
12298
Feeder F

1
1
1
1
1

July
Cluster
2
8

3

1
4
1
1
1
1
5
13

The smart meter is
336607037

8

5

Total
9
1
1
1
1
4
1
1
1
1
5
26

Transformer
1
2051
3840
3965
4611
4702
4711
11917
12187
12188
12298
Feeder F

1
1
1
1
1

June + July
Cluster
2
8

3

1
4
1
1
1
1
5
13

8

5

Total
9
1
1
1
1
4
1
1
1
1
5
26

Transformer 1: Smart Meter 336607037

MDS Visualization - Feeder D
Without compression:

With compression:

Time Stability - Feeder D

Without compression:

With compression:

July

June

A
B
C
Total

A
9

9

B
10
12
22

July

C
17
7
24

Total
36
12
7
55

June

A
B
C
Total

A
39

B

C

13
39

13

3
5

Total
39
13
5
55

Same Transformer, Same Cluster ?
Yes for Feeder D
June
Cluster
Transfor
mer
58
59
60
61
77
92
117
154
261
277
315
324
457
490
504
651
928
980
982
1033
1374
1390
1392
1409
1863
1893
12454
12565
12788
12894
13195
13197
13597
13714
13718
13799
14281
14614
15109
Feeder D

1
1
1
1
1
2
1

2

July
Cluster
3

1
1
1
4
1
1
1
1
1
1
2
1
1
1
1
3
1
2

2
1
1
1
1
1
2
1
4
1
1
1
2
3
39

1
13

3

Total
1
1
1
1
2
1
1
1
1
4
1
1
1
1
1
1
2
1
1
1
1
3
1
2
2
1
1
1
1
1
2
1
4
1
1
1
2
3
1
55

Transfor
mer
58
59
60
61
77
92
117
154
261
277
315
324
457
490
504
651
928
980
982
1033
1374
1390
1392
1409
1863
1893
12454
12565
12788
12894
13195
13197
13597
13714
13718
13799
14281
14614
15109
Feeder D

1
1
1
1
1
2
1

2

June + July
Cluster
3

1
1
1
4
1
1
1
1
1
1
2
1
1
1
1
3
1
2
2

1
1
1
1
1
2
1
4
1
1
1
2
3

39

1
13

3

Total
1
1
1
1
2
1
1
1
1
4
1
1
1
1
1
1
2
1
1
1
1
3
1
2
2
1
1
1
1
1
2
1
4
1
1
1
2
3
1
55

Transfor
mer
58
59
60
61
77
92
117
154
261
277
315
324
457
490
504
651
928
980
982
1033
1374
1390
1392
1409
1863
1893
12454
12565
12788
12894
13195
13197
13597
13714
13718
13799
14281
14614
15109
Feeder D

1
1
1
1
1
2
1

2

3

1
1
1
4
1
1
1
1
1
1
2
1
1
1
1
3
1
2
2

1
1
1
1
1
2
1
4
1
1
1
2
3

39

1
13

3

Total
1
1
1
1
2
1
1
1
1
4
1
1
1
1
1
1
2
1
1
1
1
3
1
2
2
1
1
1
1
1
2
1
4
1
1
1
2
3
1
55

Future Work
• More clustering!
• Additional feeders
• Additional time periods
• Different Fourier compressions

• Automate selection of Fourier coefficients over time
• Cluster with derivative of time series

Acknowledgements
• snəw̓eyəɬ leləm̓, Langara College is on the unceded traditional
territory of the Musqueam First Nation
• Harris SmartWorks
• IEEE
• Funding and support from Langara’s
• Post Degree Diploma program
• Work on Campus program
• Applied Research Centre

Contact
• Jeremy Chiu, Langara College
jeremychiu@langara.ca
• Albert Wong, Langara College
alwong@langara.ca

Questions?