Robust Unscented Kalman Filter for Power System Dynamic State
Estimation using PMUs
Lamine MiliProfessor, IEEE Life Fellow, [email protected]
Junbo ZhaoAssistant Professor, [email protected]
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
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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…
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Non-Gaussian PMU noise (H. Huang from PNNL)
P and Q
Voltage and
current angles
Voltage
and current
magnitudes
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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
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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
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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)
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.
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Table I Definitions of the three types of outliers
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
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Outliers
M
1j jmed l
2j
jm
ed
l
jl
1jl
2jl
1jl
2jl
Median valueProjection
Projected
values
Confidence ellipse
Projection statistic
outliers
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.
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