high-dimensional data analytics using low-dimensional models...

The First IEEE International Conference on Smart Grid Synchronized Measurements and AnalyticsTexas A&M University, College Station, Texas, USA | May 20-23, 2019

1

High-dimensional Data Analytics Using

Low-dimensional Models in Power Systems

Meng Wang

Assistant ProfessorDepartment of Electrical, Computer & Systems Engineering

Rensselaer Polytechnic Institute (RPI)Troy, NY, USA

IEEE SGSMA 2019May 20, 2019


2

Acknowledgment

Dr. Yingshuai Hao Dr. Pengzhi Gao Shuai Zhang Ren Wang

Prof. Joe H. Chow


3

Big Data and Low-Dimensional Models

Figure: Big data inpower systems Figure: Big data in

social networks

Figure: Big data in objectrecognition

• Despite the ambient dimension, many high-dimensionaldatasets have intrinsic low-dimensional structures such assparsity, low-rankness, and low-dimensional manifolds.

• These low-dimensional models enable the development of fast,model-free methods with provable performance guarantees fordata recovery and information extraction.


4

Big Data in Power Systems

• Phasor Measurement Units (PMUs)• PMUs provide synchronized phasor measurements at a

sampling rate of 30 or 60 samples per second.• Multi-channel PMUs can measure bus voltage phasors, line

current phasors, and frequency. 2000+ PMUs in the NorthAmerica.

• Data availability and quality issues, e.g., data losses due tocommunication congestions.

• Limited incorporation into the real-time operations.


5

Low Dimensionality of PMU data

Figure: PMUs in CentralNY Power Systems

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

5

10

15

20

Time(s)

Cu

rre

nt

Ma

gn

itud

e

Figure: Currentmagnitudes of PMU data

5 10 15 20 25 30 350

200

400

600

800

1000Distribution of singular value

Singular value index

Figure: Singular values ofthe PMU data matrix

• 6 PMUs measure 37 voltage/current phasors. 30samples/second for 20 seconds.

• Singular values decay significantly. Mostly close to zero.Singular values can be approximated by a sparse vector.

• Low-dimensionality also used in Chen, Xie, Kumar 2013,Dahal, King, Madani 2012 for dimensionality reduction.


6

Convert Data to Information

Objective: Develop computationally efficient data-driven methodsfor power system situational awareness.

• PMU data quality improvement: missing data recovery, baddata correction, and detection of cyber data attacks.

• Real-time event identification through machine learning.


7

Outline

1 Motivation

2 Data Recovery and Error Correction

3 Event Identification

4 Conclusions


8

PMU Data Quality Issues

• Data losses and errors resulting from communicationcongestions and device malfunction.

• California Independent System Operator reported that10%-17% of data in 2011 had availability and quality issues.

• Reliable data needed for real-time situational awareness andcontrol.


9

Simultaneous and Consecutive Data Losses

A recorded PMU dataset: consecutive data losses on three phasesof line for an hour.

00:00:00:000 04:48:00:000 09:36:00:000 14:24:00:000 19:12:00:000 24:00:00:000

Time

79

80

81

82

83

84

Mag

nitu

de (

kV)

Observed voltage magnitudes

Phase APhase BPhase C

Figure: Measured voltage phasormagnitudes

00:00:00:000 04:48:00:000 09:36:00:000 14:24:00:000 19:12:00:000 24:00:00:000

Time

0

50

100

150

Mag

nitu

de (

A)

Observed current magnitudesPhase APhase BPhase C

Figure: Measured current phasormagnitudes


10

Low-rank Matrix Completion

Low-rank Matrix Completion Problem!

• A literature includes theoretical analysis (e.g., Candes, Recht2012) and recovery methods (e.g., nuclear norm minimization(Fazel 2002)).

minX‖X‖∗ = sum of singular values of X

s.t. X is consistent with the observed entries,

• Applications in collaborative filtering, computer vision, remotesensing, load forecasting, electricity market inference


11








12








13

Low-rank Matrix Completion for PMU Data Recovery

Advantages:

• No modeling of the power system.

• Analytical performance guarantee.

• Tolerate a significant percentage ofmissing data/bad measurements atrandom locations.

y1 y2 yt

An m-by-t PMU data matrix

Time：1 ~ t

Temporally correlated

through dynamical systems

Related

through

circuit laws

ch

an

nels

: 1

~ m

Limitations:

• Do not model temporal dynamics sufficiently.

• Low-rank matrix completion methods fail to recover acolumn/row if the complete column/row is lost. Simultaneousand consecutive data losses are frequent in PMU data.

• Convex optimization problems are computationally expensivefor large datasets.


14

Low-rank Matrix Completion for PMU Data Recovery

Advantages:

• No modeling of the power system.

• Analytical performance guarantee.

• Tolerate a significant percentage ofmissing data/bad measurements atrandom locations.

y1 y2 yt


Time：1 ~ t



Related

through

circuit laws

ch

an

nels

: 1

~ m

Limitations:

• Do not model temporal dynamics sufficiently.

• Low-rank matrix completion methods fail to recover acolumn/row if the complete column/row is lost. Simultaneousand consecutive data losses are frequent in PMU data.

• Convex optimization problems are computationally expensivefor large datasets.


15

Our Contribution

Our developed model-free data recovery and error correctionmethods

• First-order algorithms to solve nonconvex optimizationproblems with provable global optimality.

• Recover/correct simultaneous and consecutive datalosses/errors.

• Differentiate bad data from system events.

Zhang, Hao, Wang, Chow. IEEE Journal of Selected Topics on Signal Processing,

2018.

Hao, Wang, Chow, Farantatos, Patel, IEEE Transactions on Power Systems, 2018.

Zhang, Wang. IEEE Transactions on Signal Processing, 2019.


16

Low-rank Hankel Structure of PMU Data

Observation matrix:

Y = [y1, y2, · · · , yn] ∈ Cnc×n

Hankel structure:

Hκ(Y) =

y1 y2 · · · yn−κ+1

y2 y3 · · · yn−κ+2...

.... . .

...yκ yκ+1 · · · yn

.Hκ(Y) ∈ Cκnc×(n−κ+1) canstill be approximated by alow-rank matrix.

y1 y2 yt


Time：1 ~ t



Related

through

circuit laws

chan

nel

s: 1

~ m

An mj-by-(t-j+1) Hankel matrix

Time

chan

nel

s

Tim

e

y1 y2

y2 y3

yj

chan

nel

sch

an

nel

s

yt-j+1

yt-j+2

ytyj+1

The low-rank Hankel property results from the reduced-orderdynamical system.


17

Low-rank Hankel Structure of PMU Data

Figure: Measurements that contain a disturbance

0 5 10 15 20 25r: approximate rank of H5(X)

10-5

10-4

10-3

10-2

10-1

100

Ap

pro

xim

atio

n e

rro

r

5=15=35=55=85=15

Figure: The low-rank approximationerrors to Hκ(Y)

0 5 10 15 20 25r: approximate rank of H5(X)

10-5

10-4

10-3

10-2

10-1

100

Ap

pro

xim

atio

n e

rro

r

5=15=35=55=85=15

Figure: The low-rank approximationerrors to Hκ(Y), where Y is acolumn permutation of Y.


18

Robust Data Recovery

Let M = Y + S denote the partially corrupted measurements,where S denotes the sparse errors.The robust data recovery problem is formulated as

minX,S∈Cnc×n

‖PΩ(X + S−M)‖2F

subject to rank(Hκ(X)) = r , ‖S‖0 ≤ s.(1)


19

Our proposed alternating projection algorithm

Initialization: X0 = 0, thresholding ε0;Two stages of iterations:

• In the k-th outer iteration:

• Increase the desired rank k from 1 to r gradually;

• In the l-th inner iteration:

• Update Sl based on the current estimated thresholding ξl ;• Update Xl along the gradient descent directionPΩ(Xl + Sl −M);

• Project the Hankel matrix HκXl into the rank-k matrix set;• Obtain Xl+1 from the matrix after projection;• Update ξl+1 based on Xl+1.


20

Theoretical results

Theorem

Suppose the number of observed data exceeds O(r3 log2(n)) andeach row of S has at most O

(1r

)fraction of nonzeros, the

algorithm converges to the original data matrix linearly as

‖Xl − Y‖F ≤ ε after l = O(log(1/ε)) iterations.

• Required number of observations: O(r3 log2(n)), less than thebound O(nr log2(n)) of recovery with convex relaxationapproach;

• Fraction of corruptions it can correct: O(

1r

)in each row;

• Low computational complexity per iteration: O(rncn log n);

• Recovery guarantees on simultaneous data losses andcorruptions across all channels.


21

Numerical experiments

0 5 10 15 20

Time (second)

0.9

0.95

1

1.05

1.1

Vol

tage

mag

nitu

de (

p.u.

) Observed voltage phasor magnitudes

0 5 10 15 20

Time (second)

-200

-100

0

100

200

Vol

tage

pha

sor

angl

es (° ) Observed angles of voltage phasors

0 5 10 15 20

Time (second)

0.9

0.95

1

1.05

1.1

Vol

tage

mag

nitu

de (

p.u.

) Recovered voltage phasor magnitudes

0 5 10 15 20

Time (second)

-200

-100

0

100

200

Vol

tage

pha

sor

angl

es (° ) Recovered voltage phasor angles

Figure: One case of 8% random bad data and 40% random missing data

0 5 10 15 20

Time (second)

0.9

0.95

1

1.05

1.1

Vol

tage

mag

nitu

de (

p.u.

) Observed voltage phasor magnitudes

0 5 10 15 20

Time (second)

0.9

0.95

1

1.05

1.1

Vol

tage

mag

nitu

de (

p.u.

) Recovered voltage phasor magnitudes

Figure: Consecutive bad data, 3% random bad data and 20% missingdata


22

Outline

1 Motivation

2 Data Recovery and Error Correction

3 Event Identification

4 Conclusions


23

Existing Approaches

Recent development of data driven approaches of eventidentification, see sample work [Wang W et al., 2014,Rafferty et al., 2016, Valtierra R et al., 2014, Jiang H et al., 2014]:

• Advantage: model-free, robust to model errors.• Limitations

• Single events or multiple events with long time intervals andminor overlapping

• A large number of training datasets• No clear physical interpretations• High training complexity.


24

Challenges and Goals

• Challenges• Events happen close in time or location have overlapping

impacts on the measurements.• Not enough training datasets for all possible conditions and

successive events.• The rapid occurrence of cascading failures requires online

algorithms.

• Goals• Train on the a small number of single events• Identify overlapping successive events in real time


25Our Approach: Identification using Subspace

Representation

• Identify an event by comparing the row subspace of thereal-time PMU data matrix with a dictionary of subspacesobtained from recorded PMU data.

M1 =

Mn1 =

...

M1 =

Mn2 =

...

M1 =

Mn3 =

...

... ... ...

Short Circuit Events

D=

Dictionary Atoms

...

...

...Line Trip Events Load Change Events

M =

Online data Offline data

Compare the

subspace of

the real-time

data with the

dictionary to

determine the

event type

Figure: Dictionary construction from historical datasets andreal-time data identification through subspace comparison

Wenting Li, Meng Wang, Joe H.Chow. IEEE Transactions on Power System, 2018.

Wenting Li, Meng Wang. To appear in IEEE Transactions on Power System, 2019.


26

Offline Training

20 40 60 80 100 120 140

Time (0.033 second)

0.96

0.97

0.98

0.99

1

The 1st layer The 2nd layer

Fully connected layer

…...

...

…...

...

Singular Values

Eigen Values

OperatorsLine Trip The output

Vector

Generator Trip

Three phase short circuit

CNN ClassifierSingle Events Training Data

Offline Training

20 40 60 80 100 120 140

Time (0.033 second)

0.9

0.92

0.94

0.96

0.98

1

1.02

20 40 60 80 100 120 140

Time (0.033 second)

0

0.2

0.4

0.6

0.8

1

1.2

Figure: Offline Training

• Extract dominant eigenvaluesλ of the state matrix and thedominant singular values δ ofthe data matrix of recordedsingle events as features;

• Input λ and σ to a 2-layerconvolutional neural network(CNN) and combine twopaths in the fully connectedlayer;

• Train this 2-layer CNNclassifier to identify the type ofan event.


27

Online Testing

Given the two successive events occurring at T1 and T2

respectively, there are three steps to identify the type of secondevent:

Prediction of the impacts

Subtraction

Prediction-Subtraction Process

Compute eigenvalues and singular values

Feature Extraction

Classify the dominant features

Classification

Two successive events

The predicted first event

The residual event

+

-

The trained CNN

classifier

The type of

the second event

Online Testing

Successive Events Testing Data

20 40 60 80 100

Time (0.033 second)

-0.08

-0.06

-0.04

-0.02

0

20 40 60 80 100

Time (0.033 second)

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

20 40 60 80 100

Time (0.033 second)

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

T2 T1

Figure: Online Testing

• Step 1: predict the impact ofthe first event after time T2

and subtract it from theobtained measurements

• Various methods can beemployed to predict themeasurements, such as timeseries analysis and Hankelmatrix based methods[Hao et al., 2018].


28

Online Testing




Subtraction



Feature Extraction


Classification



The residual event

+

-

The trained CNN

classifier

The type of

the second event

Online Testing


20 40 60 80 100

Time (0.033 second)

-0.08

-0.06

-0.04

-0.02

0

20 40 60 80 100

Time (0.033 second)

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

20 40 60 80 100

Time (0.033 second)

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

T2 T1


• Step 1: predict the impact ofthe first event after time T2

and subtract it from theobtained measurements

• Various methods can beemployed to predict themeasurements, such as timeseries analysis and Hankelmatrix based methods[Hao et al., 2018].


29

Online Testing




Subtraction



Feature Extraction


Classification



The residual event

+

-

The trained CNN

classifier

The type of

the second event

Online Testing


20 40 60 80 100

Time (0.033 second)

-0.08

-0.06

-0.04

-0.02

0

20 40 60 80 100

Time (0.033 second)

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

20 40 60 80 100

Time (0.033 second)

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

T2 T1


• Step 2: The features (λ, σ)are extracted from the residualmeasurements;

• Step 3: The trained CNNoutputs the type of thesecond event in real time.


30

Online Testing




Subtraction



Feature Extraction


Classification



The residual event

+

-

The trained CNN

classifier

The type of

the second event

Online Testing


20 40 60 80 100

Time (0.033 second)

-0.08

-0.06

-0.04

-0.02

0

20 40 60 80 100

Time (0.033 second)

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

20 40 60 80 100

Time (0.033 second)

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

T2 T1


• Step 2: The features (λ, σ)are extracted from the residualmeasurements;

• Step 3: The trained CNNoutputs the type of thesecond event in real time.


31

Numerical Results

Table: Total Number of the Testing Datasets

Types LT+GT LT+TP LT+LT GT+GT GT+TP TP+GT

Number 169 404 378 107 58 146

• Line trips (LTs), generator trips (GTs), and three-phase shortcircuit (TPs) in the 68-bus power system generated usingPSS/E;

• The training set includes 967 single events at differentlocations with different topologies and various pre-eventconditions;

• The testing set includes 1262 two-event cases, where“LT+GT” means a line trip event is followed by a generatortrip event.

• The time interval between any two successive events variesfrom 0.5 to 2 seconds.


32

Performance with a Small Training Dataset

6 12 18 24 30 50 80 100

Percentage of Training Data (%)

40

60

80

100

IAR

(%

)

LT

6 12 18 24 30 50 80 100


40

60

80

100

IAR

(%

)

GT

6 12 18 24 30 50 80 100


40

60

80

100

IAR

(%

)

TP

6 12 18 24 30 50 80 100


40

60

80

100

IAR

(%

)

Averaged

CNN-F CNN-T

• CNN-F achieves a higher IAR when given a small trainingdata;

• The IAR of CNN-T is sensitive to the size of the trainingdata.


33

The Impact of Subtracting the First Event

Table: IAR of CNN-F and CNN-T with and without subtracting the firstevent (∆T = 1 second)

Classifier Process LT % GT % TP % Overall %

CNN-F NS 79.4 68.3 96.4 81.9

CNN-F SP 95.9 89.1 97.3 94.2

CNN-T NS 94.8 83.2 78.4 85.1

CNN-T SP 73.2 62.4 78.4 71.5

• “Not Subtract (NS)” means using the measurements after thesecond event directly;

• “Subtract the Prediction (SP)” means using the residual aftersubtracting the first event;

• Subtracting the impact of the first event can enhanceCNN-F’s performance significantly.


34

Robustness to Noise

Table: IAR of identifying the second event by CNN-F with differentsignal-noise-ratio (SNR) noisy measurements

SNR (dB) 40 50 60 70 80 90 100

IAR of LT (%) 77.3 81.4 82.5 85.6 82.5 90.7 91.7

IAR of GT (%) 75.2 82.2 86.1 86.1 85.1 86.1 82.2

IAR of TP (%) 93.7 97.3 97.3 98.2 98.2 99.1 99.1

Overall IAR (%) 82.5 87.4 88.9 90.3 90.0 92.2 91.3

• The overall IAR of CNN-F is more than 80% for differentnoise levels;

• IAR is more than 90% when the SNR is higher than 70 dB;

• Significant events like TP events are less sensitive to noise.


35

Conclusions

• A framework of power system data analytics by exploiting thelow-dimensional structure of spatial-temporal data blocks.

• Data quality improvement with analytical guarantees.(Missing data recovery, detection of cyber data attacks.)

• Real-time event identification approach using a small numberof recorded single events for training.


36

Q & A


37

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