Tao Jiang, Ion Matei and John S. Baras Institute for Systems Research

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 ta 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 cossin sin

x a x ax a x ax 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 tc t c t tc 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 ky 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 i

j

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

G3G4

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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

in the distributed Kalman filter algorithm

Conclusions

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Thank you!

baras@isr.umd.edu301-405-6606

http://www.isr.umd.edu/~baras

Questions?