Robust Unscented Kalman Filter for Power System Dynamic State

Estimation using PMUs

Lamine MiliProfessor, IEEE Life Fellow, lmili@vt.edu

Junbo ZhaoAssistant Professor, zjunbo@vt.edu

Bradley Department of Electrical and Computer EngineeringVirginia Tech, Northern Virginia Center

Falls Church, VA 22043, USA

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2

Dynamic Security Analysis: Perform real-time state prediction and filtering for rotor angle and voltage stability.

Dynamic State Estimation- Motivations

Power System Dynamics

Dynamic State Estimation

Continuous Generator

Feedback Controls

Discontinuous Controls(Trip loads or generators, etc.)

Unstable

Stable

Rotor Angle and Voltage Stability

Analysis

3

Why Robust DSE for Power Systems?PMU data quality issues:

• Large measurement bias

• Non-Gaussian noise

• Data dropout/packet loss

• Loss of GPS synchronization

• Measurement delays

• Bad/missing timestamps

• Cyber attack…

4

Non-Gaussian PMU noise (H. Huang from PNNL)

P and Q

Voltage and

current angles

Voltage

and current

magnitudes

5

Occurrence of Outliers (PMU Data from Brazil Utility)

0 50 100 150 200-0.5

0

0.5

1

1.5

2

Time [seconds]

An

gle

[ra

dia

ns]

UFAM

UFMA

UFPA

UFRGS

UFSC

UNIPAMPA

Bad data

Event

Brazil utility

6

How Does the KF Work?

Update Stage

Prediction Stage

1|

1|1||

1

1|1|

)(

]ˆ[ˆˆ

][

kkkkkk

kkkkkkkkk

T

k

T

kkkk

T

kkkk

HKI

xHzKxx

RHHHKCompute the filter gain

Correct the prediction

Update the covariance

6

k

T

kkkkkk

kkkkk

WFF

xFx

11|111|

1|111|

ˆ ˆPredict the state

Compute covariance

System

process

noise

covariance

matrix

Observation noise

covariance matrix

Problem Formulation7

Dynamical system discrete-time state-space model:

𝔼 𝒘𝑘−1𝒘𝑘−1𝑇 = 𝑸𝑘

𝔼 𝒗𝑘𝒗𝑘𝑇 = 𝑹𝑘

𝒙𝑘 = 𝒇 𝒙𝑘−1 +𝒘𝑘−1

𝒛𝑘 = 𝒉 𝒙𝑘−1 + 𝒗𝑘

“State Prediction”Dynamic Simulation

“State correction”

Prediction step Filtering step

𝒙𝑘 = 𝒇 𝒙𝑘−1 +𝒘𝑘−1𝒛𝑘 = 𝒉 𝒙𝑘−1 + 𝒗𝑘

𝒛𝑘

𝒙𝑘−1 𝒙𝑘

Dynamic state estimation framework:

(1)

Extended Kalman filter (EKF) and Unscented Kalman filter (UKF)

8

Drawbacks of EKF and UKF• They are based on Least Squares Estimator and thus

highly sensitive to deviations from the assumptions;

• They provide biased results with non-Gaussian noise;

• Lack of robustness to any type of outliers: observation, innovation and structural outliers;

• Sensitive to cyber attacks, measurement losses, etc.

9

Table I Definitions of the three types of outliers

Proposed Robust GM-UKF

Batch-Mode Regression FormCombine the measurement equation and the prediction equation to obtain the batch-mode regression form :

Prediction equation:

𝒙𝑘 = 𝒙𝑘|𝑘−1 + 𝜻𝑘|𝑘−1

Measurement equation: Perform statistical linearization of𝒉(𝒙𝑘) around the predicted state 𝒙𝑘|𝑘−1:

𝒛𝑘 = 𝒉 𝒙𝑘|𝑘−1 +𝑯𝑘 𝒙𝑘 − 𝒙𝑘|𝑘−1 + 𝝊𝑘 + 𝒆𝑘

where 𝑯𝑘 = 𝑷𝑘|𝑘−1𝑥𝑧 𝑇

𝑷𝑘|𝑘−1𝑥𝑥 −1

and 𝒆𝑘 is the statistical

linearization error.

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Predicted state

True value

Prediction error

(2)

(3)

Batch-mode regression form

𝒛𝑘 − 𝒉 𝒙𝑘|𝑘−1 +𝑯𝑘 𝒙𝑘|𝑘−1 𝒙𝑘|𝑘−1

= 𝑯𝑘𝑰𝒙𝑘 +𝝊𝑘 + 𝒆𝑘−𝜻𝑘|𝑘−1

𝒛𝑘 = 𝑯𝑘 + 𝒆𝑘

• The covariance matrix of the error 𝒆𝑘 is given by

𝔼 𝒆𝑘 𝒆𝑘𝑇 =𝑹𝑘 + 𝑹𝑘 00 𝜮𝑘|𝑘−1

= 𝑺𝑘𝑺𝑘𝑇 ,

where 𝔼 𝒆𝑘𝒆𝑘𝑇 = 𝑹𝑘 and 𝑺𝑘is obtained from

Cholesky decomposition and used for prewhitening after outlier detection.

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(4)

(5)

Detecting Outliers by PS

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Scatter of matrix Z without outliers Scatter of matrix Z with outliers

w

d and Innovation

P, Q, V

Projection statistic

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111 12

21 22 2

1 2

...

T

T

Tm m m

zz z

z z zZ

z z z

K K M

1

( )

max1.4826 ( )

T T

i j

j

i TT

jkk

z med z

PSmed z med z

15

Outliers

M

1j jmed l

2j

jm

ed

l

jl

1jl

2jl

1jl

2jl

Median valueProjection

Projected

values

Confidence ellipse

Projection statistic

outliers

The Weight Function based on PS

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d2

1

𝜛𝑖

2iPS

𝜛𝑖 = min{1,𝑑

𝑃𝑆𝑖

2}

where 𝑑 c22,0.975

-d2

(8)

Robust PrewhiteningRemember:

𝔼 𝒆𝑘 𝒆𝑘𝑇 =𝑹𝑘 + 𝑹𝑘 00 𝜮𝑘|𝑘−1

= 𝑺𝑘𝑺𝑘𝑇

𝑺𝒌 is robust because:

• 𝜮𝒌|𝒌−𝟏 is robust thanks to the robust covariance matrix

updating.

Finally:

𝑺𝑘−1 𝒛𝑘 = 𝑺𝑘

−1 𝑯𝑘 𝒙𝑘 + 𝑺𝑘−1 𝒆𝑘

𝒚𝑘 = 𝑨𝑘𝒙𝑘 + 𝝃𝑘

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(9)

(10)

Robust GM-estimator

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A GM-estimator minimizes an objective function given by

arg𝒎𝒊𝒏 𝑱 𝒙 = 𝒊=𝟏𝒎 𝝕𝒊𝟐 𝝆 𝒓𝑺𝒊

Convex Huber 𝝆-function

Robust Filtering

𝝏𝑱 𝒙

𝝏𝒙=

𝒊=𝟏

𝒎

−𝝕𝒊𝒂𝒊𝒔𝝍 𝒓𝑺𝒊 = 𝟎

• The minimization problem is solved via the Iteratively Reweighted Least Squares (IRLS) algorithm:

𝒙𝒌|𝒌(𝒋+𝟏)= 𝑨𝒌𝑻𝑸(𝒋)𝑨𝒌

−𝟏𝑨𝒌𝑻𝑸(𝒋)𝒚𝒌

𝑸 = diag 𝒒 𝒓𝑺𝒊 and 𝒒 𝒓𝑺𝒊 = 𝝍 𝒓𝑺𝒊 𝒓𝑺𝒊.

• Stopping rule:

𝒙𝒌|𝒌(𝒋+𝟏)− 𝒙𝒌|𝒌(𝒋)< 𝟏𝟎−𝟐

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(14)

(15)

(16)

Robust covariance matrix updatingThe estimated state by our GM-UKF tends to a Gaussian distribution asymptotically even when the system process and measurement noise follow a non-Gaussian distribution. Furthermore, the estimation error covariance matrix is updated through

𝜮𝑘|𝑘 = 1.0369 𝑨𝑘𝑇𝑨𝑘−1𝑨𝑘𝑇𝑸𝜛𝑨𝑘

−1𝑨𝑘𝑇𝑨𝑘−1

where𝑸𝝕 = diag 𝜛𝒊𝟐 .

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𝑸𝜛 determined by PS is used to downweight outliers, yielding robust covariance matrix updating.

(17)

Disturbance: at t=0.5 seconds, transmission line between buses 15 and 16 is removed.

Generator model: two-axis model with IEEE DC1A excitation system and TGOV1 turbine-governor is assumed.

Non-Gaussian noise:

Bimodal Gaussian mixture noise with zero mean, variances of 10−4 and 10−3and weights of 0.9 and 0.1, respectively, is added to the voltage magnitudes;

Laplacian noise with zero mean and scale 0.2 is added to the real and reactive power injections.

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Illustrative Results on IEEE 39-bus system

Case 1: Non-Gaussian Noise•No outliers;

•Bimodal Gaussian mixture for current and voltage magnitudes;

•Laplace noises for real and reactive power;

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Case 1: Non-Gaussian Noise

Case 2: Observation Outliers•The real and reactive power measurements of Generator 5 are

corrupted with 20% error from 4s to 6s; Laplace noises for real

and reactive power.

• State estimates by UKF

are significantly biased;

• GM-UKF achieves

much higher statistical

efficiency than Huber-

UKF and GM-IEKF.

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Case 3: Parameter Errors•The predicted rotor angle of the Generator 5 is incorrect due to

the incorrect parameter of G5 from 4s to 6s; Laplace noises for

real and reactive power.

•State estimates by UKF

and Huber-UKF are

significantly;

•GM-UKF achieves much

higher statistical efficiency

than GM-IEKF.

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Table II. Average Computing Time at Each PMU Sample (PC with Intel Core i5, 2.50 GHz, 8GB of RAM)

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Cases EKF UKF GM-IEKF GM-UKF

Case 1 6.24ms 6.28ms 9.64ms 9.52ms

Case 2 6.28ms 6.31ms 9.68ms 9.55ms

Case 3 6.43ms 6.38ms 9.72ms 9.63ms

Case 4 6.45ms 6.40ms 9.71ms 9.62ms

Case 5 6.25ms 6.29ms 9.66ms 9.54ms

Breakdown point and computing efficiencyHandle at least 25% outliers due to cyber attacks, PMU

communication issues or model deficiency;

Suitable for real-time application.

ConclusionsGM-UKF is able to track power system dynamics even

when system nonlinearities are strong;

GM-UKF exhibits high statistical efficiency under non-Gaussian noise while being able to suppress observation and innovation outliers;

It has high breakdown point (25%) to cyber attacks and model deficiency.

Future work will be the development of hybrid DSE that integrates GM-UKF with robust control theory to address model uncertainties.

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Reference27

J. B. Zhao, L. Mili, ``A Robust Generalized-Maximum Likelihood Unscented Kalman Filter for Power System Dynamic State Estimation,” IEEE Journal of Selected Topics in Signal Processing, 2018.

J. B. Zhao, L. Mili, "Power System Robust Decentralized Dynamic State Estimation Based on Multiple Hypothesis Testing," IEEE Trans. on Power Systems, DOI: 10.1109/TPWRS.2017.2785344, 2017, in press.

J. B. Zhao, L. Mili, "Robust Unscented Kalman Filter for Power System Dynamic State Estimation with Unknown Noise Statistics," IEEE Trans. on Smart Grid, DOI: 10.1109/TSG.2017.2761452, 2017.

S. Wang, J. B. Zhao, Z. Huang, R. Diao, "Assessing Gaussian Assumption of PMU Measurement Error Using Field Data," IEEE Trans. on Power Delivery, DOI: 10.1109/TPWRD.2017.2762927, 2017, in press.

J. B. Zhao, M. Netto, L. Mili, "A Robust Iterated Extended Kalman Filter for Power System Dynamic State Estimation", IEEE Trans. on Power Systems, vol. 32, no. 4, pp. 3205-3216, 2017.

J. B. Zhao, L. Mili, F. Milano, "Robust Frequency Divider for Power System Online Monitoring and Controls," IEEE Trans. on Power Systems, DOI: 10.1109/TPWRS.2017.2785348, 2017, in press.

Turbine-Generator System

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Turbine Generator

Exciter

Governor

Power System

Two-axis Machine Model

𝑻𝒅𝒐′𝒅𝑬𝒒′

𝒅𝒕= −𝑬𝒒′ − 𝑿𝒅 − 𝑿𝒅

′ 𝑰𝒅 + 𝑬𝒇𝒅

𝑻𝒒𝒐′𝒅𝑬𝒅′

𝒅𝒕= −𝑬𝒅′ + 𝑿𝒒 − 𝑿𝒒

′ 𝑰𝒒

𝒅𝜹

𝒅𝒕= 𝝎 −𝝎𝒔

𝟐𝑯

𝝎𝒔

𝒅𝝎

𝒅𝒕= 𝑻𝑴 − 𝑬𝒅

′ 𝑰𝒅 − 𝑬𝒒′ 𝑰𝒒 − 𝑿𝒒

′ − 𝑿𝒅′ 𝑰𝒅𝑰𝒒 − 𝑻𝑭𝑾

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Field

winding

Damper

winding in

the q axis

Swing equations

IEEE DC1A excitation system model

𝑻𝑬𝒅𝑬𝒇𝒅

𝒅𝒕= − 𝑲𝑬 + 𝑺𝑬 𝑬𝒇𝒅 𝑬𝒇𝒅 + 𝑽𝑹

𝑻𝑭𝒅𝑹𝒇

𝒅𝒕= −𝑹𝒇 +

𝑲𝑭𝑻𝑭𝑬𝒇𝒅

𝑻𝑨𝒅𝑽𝑹𝒅𝒕= −𝑽𝑹 +𝑲𝑨𝑹𝒇 −

𝑲𝑨𝑲𝑭𝑻𝑭𝑬𝒇𝒅 +𝑲𝑨 𝑽𝒓𝒆𝒇 − 𝑽𝒕

30

𝑻𝑪𝑯𝒅𝑻𝑴𝒅𝒕= −𝑻𝑴 + 𝑷𝑺𝑽

𝑻𝑺𝑽𝒅𝑷𝑺𝑽𝒅𝒕= −𝑷𝑺𝑽 + 𝑷𝑪 −

𝟏

𝑹𝑫

𝝎

𝝎𝒔− 𝟏

TGOV1 turbine-governor model