Robust Dynamic State Estimation for PMU Data Quality and Security

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IEEE PES GM 2019 Panel

Dynamic State Estimation for Power System Monitoring, Protection and Control--Paving the Way for A More Resilient Grid

Assistant ProfessorDepartment of Electrical and Computer Engineering

Mississippi State UniversityStarkville, MS, 39762, USA

Junbo Zhao, Ph.D.

Acknowledgement-Professor Lamine Mili

Outline

Introduction of Power System DSE

Motivations and Implementations

Robust DSE Needs

Proposed Robust Unscented Kalman Filter

Bad Data and Cyber Attack Detection

Robust Filtering

Results and Discussions

Conclusions

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Source: Solar in US-NREL

Integration of intermittent DERs and flexible loads adds more uncertainties and dynamics to the system operating states;Steady-state assumption may be violated.

Source: EV in US-IHS Automotive

Dynamic State Estimation- Motivations3

Synchrophasor measurements at 30-60 samples/s, provide the opportunity to capture power system dynamics.

Source: www.naspi.org

Dynamic State Estimation- Motivations4

Dynamic State Estimator

Exciter

Governor

Synchronous Generators with PMU

Measurements

P

V

refV

MP

fdE

refP

'

dE '

qEAdvanced System

Control and Protection

For advanced protection and control of power systems.

Q

Dynamic State Estimation- Motivations5

Discrete-time state-space model of the power system DAE:

𝔼 𝒘𝑘𝒘𝑘𝑇 = 𝑸𝑘

𝔼 𝒗𝑘𝒗𝑘𝑇 = 𝑹𝑘

𝒙𝑘 = 𝒇 𝒙𝑘−1, 𝒚𝑘−1, 𝒖𝑘 +𝒘𝑘

𝒛𝑘 = 𝒉 𝒙𝑘 , 𝒚𝑘 , 𝒖𝑘 + 𝒗𝑘

subject to various types of constraints

(1)

Problem Formulation-DSE

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• Implementation using Kalman Filter Framework

Implementation of DSE

“State Prediction”Dynamic Simulation

“State correction”

Prediction step Filtering step

𝒙𝑘 = 𝒇 𝒙𝑘−1, 𝒚𝑘−1 +𝒘𝑘−1𝒛𝑘 = 𝒉 𝒙𝑘, 𝒚𝑘 + 𝒗𝑘

𝒛𝑘

𝒙𝑘−1 𝒙𝑘

Extended Kalman filter-poor performance for strong nonlinear system

Unscented Kalman filter and its variants, Ensemble Kalman filter-sensitive to non-Gaussian noise

Particle filter and its variants-sensitive to bad data

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Robust DSE-Motivations

Large measurement bias

Non-Gaussian noise

Data dropout/packet loss/measurement delays

Loss of GPS synchronization

Unknown inputs and noise statistics

Cyber attacks

PMU data quality issues:

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Non-Gaussian PMU Noise Distributions

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Voltage and current angle errors-roughly Gaussian; Voltage and current magnitude errors-Gaussian mixture; Real and reactive power errors-thick-tail distributions, such as

Laplace and Cauchy.

S. Wang, J. B. Zhao, Z. Huang, R. Diao "Assessing Gaussian Assumption of PMU Measurement

Error Using Field Data," IEEE Trans. on Power Delivery, vol. 33, no. 6, pp. 3233-3236, 2018.

Non-Gaussian PMU Noise Distributions

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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

Occurrence of Outliers (Field Data from Brazil)

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Vulnerabilities to Cyber Attacks

Proposed Robust GM-UKF

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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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Projection statistics is defined and calculated by

(6)

Outlier detection using PSWe propose to apply PS on the following matrix

𝒁 =𝒛𝑘−1 − 𝒉 𝒙𝑘−1|𝑘−2 𝒙𝑘−1|𝑘−2

𝒛𝑘 − 𝒉 𝒙𝑘|𝑘−1 𝒙𝑘|𝑘−1

Time instants: 𝑘 − 1 𝑘

Global redundancy for outlier detection is increased thanks to the batch-mode formulation.

𝒁 has two columns to measure temporal correlations. It is found two continuous samples are sufficient to detect outliers effectively.

PS is applied to the innovation vectors and the predicted state vector separately.

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Innovation vectors

Probability Distribution of PS

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Q-Q plots of the sample quantiles of the PS vs. the correspondingquantiles of the chi-square distributions with different degree offreedoms, where the left and the right figures represent Q-Q plots ofPS with Gaussian and Laplace noise, respectively.

Robust Filtering

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𝜛𝑖 = min{1,c22,0.975

𝑃𝑆𝑖

2

} (7)

arg𝑚𝑖𝑛 𝐽 𝒙 = 𝑖=1𝑚 𝜛𝑖2 𝜌 𝑟𝑆𝑖

New weighting function:

Robust filtering via the GM-estimator that minimizes an objective function given by

(8)

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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(9)

(10)

(11)

Robust Covariance Matrix UpdatingTheorem 1: The 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.

(12)

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;

Case 1: Non-Gaussian Noise23

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 Efficiency

Handle at least 25% outliers due to cyber attacks, PMU

communication issues or model deficiency;

Suitable for real-time application.

Conclusions

• DSE is excepted to be a key tool in future cyber-physical energy system EMS

• GM-UKF is able to handle non-Gaussian noise, bad data and cyber attacks while achieving good statistical efficiency

• Numerical and statistical robustness of DSE should be extensively studied

• Field validation of the developed GM-UKF will be our future work

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• J. B. Zhao, L. Mili, "A Theoretical Framework of Robust H-infinity Unscented Kalman Filter and Its Application to Power System Dynamic State Estimation," IEEE Trans. Signal Processing, vol. 67, no. 10, pp. 2734-2746, 2019.

• J. B. Zhao, A. Exposito, M. Netto, L. Mili, A. Abur, V. Terzija, I. Kamwa, B. Pal, A. K. Singh, J. Qi, Z. Huang, A. P. Sakis Meliopoulos, ''Power System Dynamic State Estimation: Motivations, Definitions, Methodologies and Future Work," IEEE Trans. Power Systems, 2019.

• 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, vol. 12, no. 4, pp. 578-592, 2018.

References

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