Robust Dynamic State Estimation for PMU Data Quality and ...
Transcript of Robust Dynamic State Estimation for PMU Data Quality and ...
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
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dE '
qEAdvanced System
Control and Protection
For advanced protection and control of power systems.
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Dynamic State Estimation- Motivations5
Discrete-time state-space model of the power system DAE:
๐ผ ๐๐๐๐๐ = ๐ธ๐
๐ผ ๐๐๐๐๐ = ๐น๐
๐๐ = ๐ ๐๐โ1, ๐๐โ1, ๐๐ +๐๐
๐๐ = ๐ ๐๐ , ๐๐ , ๐๐ + ๐๐
subject to various types of constraints
(1)
Problem Formulation-DSE
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โข Implementation using Kalman Filter Framework
Implementation of DSE
โState PredictionโDynamic Simulation
โState correctionโ
Prediction step Filtering step
๐๐ = ๐ ๐๐โ1, ๐๐โ1 +๐๐โ1๐๐ = ๐ ๐๐, ๐๐ + ๐๐
๐๐
๐๐โ1 ๐๐
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:
๐๐ = ๐๐|๐โ1 + ๐ป๐|๐โ1
Measurement equation: Perform statistical linearization of๐(๐๐) around the predicted state ๐๐|๐โ1:
๐๐ = ๐ ๐๐|๐โ1 +๐ฏ๐ ๐๐ โ ๐๐|๐โ1 + ๐๐ + ๐๐
where ๐ฏ๐ = ๐ท๐|๐โ1๐ฅ๐ง ๐
๐ท๐|๐โ1๐ฅ๐ฅ โ1
and ๐๐ is the statistical
linearization error.
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Predicted state
True value
Prediction error
(2)
(3)
Batch-mode Regression Form
๐๐ โ ๐ ๐๐|๐โ1 +๐ฏ๐ ๐๐|๐โ1 ๐๐|๐โ1
= ๐ฏ๐๐ฐ๐๐ +๐๐ + ๐๐โ๐ป๐|๐โ1
๐๐ = ๐ฏ๐ + ๐๐
โข The covariance matrix of the error ๐๐ is given by
๐ผ ๐๐ ๐๐๐ =๐น๐ + ๐น๐ 00 ๐ฎ๐|๐โ1
= ๐บ๐๐บ๐๐ ,
where ๐ผ ๐๐๐๐๐ = ๐น๐ and ๐บ๐is 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
๐ =๐๐โ1 โ ๐ ๐๐โ1|๐โ2 ๐๐โ1|๐โ2
๐๐ โ ๐ ๐๐|๐โ1 ๐๐|๐โ1
Time instants: ๐ โ 1 ๐
Global redundancy for outlier detection is increased thanks to the batch-mode formulation.
๐ 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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๐๐ = min{1,c22,0.975
๐๐๐
2
} (7)
arg๐๐๐ ๐ฝ ๐ = ๐=1๐ ๐๐2 ๐ ๐๐๐
New weighting function:
Robust filtering via the GM-estimator that minimizes an objective function given by
(8)
Robust Filtering
๐๐ฑ ๐
๐๐=
๐=๐
๐
โ๐๐๐๐๐๐ ๐๐บ๐ = ๐
โข The minimization problem is solved via the Iteratively Reweighted Least Squares (IRLS) algorithm:
๐๐|๐(๐+๐)= ๐จ๐๐ป๐ธ(๐)๐จ๐
โ๐๐จ๐๐ป๐ธ(๐)๐๐
๐ธ = diag ๐ ๐๐บ๐ and ๐ ๐๐บ๐ = ๐ ๐๐บ๐ ๐๐บ๐.
โข Stopping rule:
๐๐|๐(๐+๐)โ ๐๐|๐(๐)< ๐๐โ๐
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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
๐ฎ๐|๐ = 1.0369 ๐จ๐๐๐จ๐โ1๐จ๐๐๐ธ๐๐จ๐
โ1๐จ๐๐๐จ๐โ1
where๐ธ๐ = diag ๐๐๐ .
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๐ธ๐ 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 10โ4 and 10โ3and 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 NoiseโขNo outliers;
โขBimodal Gaussian mixture for current and voltage magnitudes;
โขLaplace noises for real and reactive power;
Case 1: Non-Gaussian Noise23
Case 2: Observation OutliersโขThe real and reactive power measurements of Generator 5 are
corrupted with 20% error from 4s to 6s; Laplace noises for real
and reactive power.
โข State estimates by UKF
are significantly biased;
โข GM-UKF achieves
much higher statistical
efficiency than Huber-
UKF and GM-IEKF.
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Case 3: Parameter ErrorsโขThe 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.
โขState estimates by UKF
and Huber-UKF are
significantly;
โขGM-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
โข DSE is excepted to be a key tool in future cyber-physical energy system EMS
โข GM-UKF is able to handle non-Gaussian noise, bad data and cyber attacks while achieving good statistical efficiency
โข Numerical and statistical robustness of DSE should be extensively studied
โข Field validation of the developed GM-UKF will be our future work
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โข 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.
โข 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.
โข 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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