A Trust Based Distributed Kalman Filtering Approach for Mode Estimation in Power Systems Tao Jiang, Ion Matei and John S. Baras Institute for Systems Research and Department of Electrical and Computer Engineering University of Maryland College Park, USA {tjiang, imatei, bara}@umd.edu The First Workshop on Secure Control Systems (SCS) Stockholm, Sweden, April 12, 2010
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A Trust Based Distributed Kalman Filtering Approach for Mode Estimation in Power Systems Tao Jiang, Ion Matei and John S. Baras Institute for Systems Research.
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A Trust Based Distributed Kalman Filtering
Approach for Mode Estimation in Power Systems
Tao Jiang, Ion Matei and John S. BarasInstitute for Systems Research
and Department of Electrical and Computer EngineeringUniversity of Maryland College Park, USA
{tjiang, imatei, bara}@umd.edu
The First Workshop on Secure Control Systems (SCS) Stockholm, Sweden, April 12, 2010
Acknowledgments
Sponsors: Research partially supported by the Defense Advanced Research Projects Agency (DARPA) under award number 013641-001 for the Multi-Scale Systems Center (MuSyC), through the Focused Research Centers Program of SRC and DARPA.
Useful discussions and suggestions received
through participation in the EU project VIKING
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Outline
Introduction Problem formulation Distributed Kalman filtering with
trust dependent weights Simulations Conclusions
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Introduction
Control and protection of power systems: Large-scale interconnected power networks Huge amount of data collection in real-time Distributed communication and control
New security requirements besides confidentiality, integrity and availability
Quality of collected data from various substations: uncertainty of data accuracy
Behavior of participants in the power grid operations: malicious, selfish
In this paper, we propose a trust based distributed Kalman filtering approach to estimate the modes of power systems.
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Introduction Problem formulation Distributed Kalman filtering with
trust dependent weights Simulations Conclusions
Problem Formulation
Inter-area oscillations (modes) Associated with large inter-connected power
networks between clusters of generators Critical in system stability Requiring on-line observation and control
Automatic estimation of modes Using currents, voltages and angle differences
measured by PMUs (Power Management Units) that are distributed throughout the power system
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Linearization
Linearization around the nominal operating points The initial steady–state value is eliminated Disturbance inputs consist of M frequency modes
defined as
oscillation amplitudes; damping constants; oscillation frequencies; phase angles of the
oscillations Consider two modes and use the first two terms in the
Taylor series expansion of the exponential function; expanding the trigonometric functions:
1 1 12
( ) exp( )cos( ) exp( )cos( )M
i i i ii
f t a t t a t t
:ia :i:i :i
1 1 1
2 2 2 2 2 2
( ) (1 )cos( )
(1 )[cos cos( ) sin sin( )].
f t a t t
a t t t
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Linearization (cont’)
Introducing the notation:
where j stands for the measurement j The power system is sampled at a preselected rate,
then we have the discrete-time linear measurement model
vj (k) is the measurement noise assumed Gaussian with zero mean and covariance matrix Rj
1 1 2 1 1
3 2 2 4 2 2 2
5 2 2 6 2 2 2
cos cos
sin sin
x a x a
x a x a
x a x a
1 1 2 1
3 2 4 2
5 2 6 2
cos( ) cos( )
cos( ) cos( )
sin( ) sin( )
j j
j j
j j
c t c t t
c t c t t
c t c t
( ) ( ) ( ) ( )j j jy k C k x k v k
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Linear System Model
Assume N measurements by N PMUs and define A(k) as the identity matrix
w(k) is the state noise assumed Gaussian with zero mean and covariance matrix Q
The initial state x0 is assumed to be a Gaussian distribution with mean μ0 and covariance matrix P0
The linear random process can be estimated using the Kalman filter algorithm
Having estimated the parameter vector x (k), the amplitude, damping constant, and phase angle can be calculated at any time step k
( 1) ( ) ( ) ( )
( ) ( ) ( ) ( )
x k A k x k w k
y k C k y k v k
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Distributed Estimation
To compute an accurate estimate of the state x (k), using:
local measurements yj (k); information received from the PMUs in its communication
neighborhood; confidence in the information received from other PMUs
provided by the trust model
PMUPMU
PMU
GPS Satellite
N multiple recording sites (PMUs) to measure the output signals
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Trust Model
To each information flow (link) j i, we attach a positive value Tij , which represents the trust PMU i has in the information received from PMU j ;
Trust interpretation: Accuracy Reliability
Goal: Each PMU has to compute accurate estimates of the state, by intelligently combining the measurements and the information from neighboring PMUs
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Introduction Problem formulation Distributed Kalman filtering with
trust dependent weights Simulations Conclusions
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Distributed Kalman Filtering with Trust Dependent Weights
We use for distributed state estimation -- a simplified version of an algorithm introduced in (Olfati-Saber, 2007)
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Distributed Kalman Filtering with Accuracy Dependent Consensus Step
We define the trust value Tij in terms of the estimation error given by the standard Kalman filter:
Remark: Although Mi is not the true covariance of the estimation error, it reflects the observability (through Ci ) and accuracy (through Ri ) of the PMU i
Assumption: (A, Ci ) detectable
1,
( )ij ij
T jTrace M
N
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Distributed Estimation with Reliability Dependent Consensus
Step
We assume some PMUs may send false information due to malfunctions or attacks;
Update mechanism for Tij is based on belief divergence (Kerchove, 2007), which shows how far a received estimate is from the other received estimates:
where Ni is the number of neighbors of PMU i
21 ˆ ˆ1
i
ij j lli
d x xN N
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Distributed Estimation with Reliability Dependent Consensus
Step
Compute the trust values according to:
where
Normalized trust values
if
Consensus weights
,ij i ij iT c d j N
max{ | }i ij ic d j N
,
i
ijij
ill
Tp
T
N
minij ip p 0ijT
i
ijij
ill
Tw
T
N
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Distributed Estimation with Reliability Dependent Consensus
Step
,ij i ij iT c d j N
max{ | }i ij ic d j N
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Introduction Problem formulation Distributed Kalman filtering with
trust dependent weights Simulations Conclusions
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Data from a practical example (Lee and Poon, 1990), which has two modes at ω1=0.4Hz and ω2 = 0.5Hz.
The power system model employs five measurements, where each PMU is installed over a line connected to one generator
Simulations
G1
G5G2
G3
G4
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Distributed Kalman Filtering with accuracy dependent consensus step
White noise with different SNR was added to each measurement
Simulations
estimating parameter a1
estimating parameter σ1In Alg 2, larger weight is given to information coming from
PMUs with small variance of the estimation error
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estimating parameter σ1
estimating parameter a1
Distributed estimation with reliability dependentconsensus step
PMU connecting to G3 sends false information
Simulations
Alg 3 detects the false data and eliminates them from estimation; False data have influence on how fast the estimates converge
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Mode estimation in power systems is modeled as estimation of a linear random process
Two modified Distributed Kalman Filtering algorithms, which incorporate the notion of trust, are proposed
Two interpretations of trust were used: Accuracy: update scheme for the trust values based on
the estimation error Reliability: belief divergence metric and a thresholding
scheme to compute the trust values The normalized trust values were used as weights