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

1. A Trust Based Distributed Kalman FilteringApproach 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

2. 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

3. Outline • Introduction • Problem formulation • Distributed Kalman filtering with trust dependent weights • Simulations • Conclusions

4. Introduction • Control and protection of power systems: • Large-scale interconnected power networks • Huge amount of data collection in real-time • Distributedcommunication 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.

5. Introduction • Problem formulation • Distributed Kalman filtering with trust dependent weights • Simulations • Conclusions

6. 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

7. 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:

8. 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

9. 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

10. Distributed Estimation GPS Satellite • 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 • N multiple recording sites (PMUs) to measurethe output signals PMU PMU PMU

11. 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

12. Introduction • Problem formulation • Distributed Kalman filtering with trust dependent weights • Simulations • Conclusions

13. Distributed KalmanFiltering with Trust Dependent Weights • We use for distributed state estimation -- a simplified version of an algorithm introduced in (Olfati-Saber, 2007)

14. Distributed Kalman Filtering with AccuracyDependent Consensus Step • We define the trust value Tij in terms of the estimation error given by the standard Kalman filter: • Remark:Although Miis 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

15. Distributed Estimation with Reliability DependentConsensus 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 Niis the number of neighbors of PMU i

16. Distributed Estimation with Reliability DependentConsensus Step • Compute the trust values according to: where • Normalized trust values if • Consensus weights

17. Distributed Estimation with Reliability DependentConsensus Step

18. Introduction • Problem formulation • Distributed Kalman filtering with trust dependent weights • Simulations • Conclusions

19. Simulations • 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 G1 G4 G3 G5 G2

20. Simulations • Distributed Kalman Filtering with accuracy dependent consensus step • White noise with different SNR was added to each measurement estimating parameter a1 estimating parameter σ1 In Alg 2, larger weight is given to information coming from PMUs with small variance of the estimation error

21. Simulations • Distributed estimation with reliability dependentconsensus step • PMU connecting to G3 sends false information estimating parameter a1 estimating parameter σ1 Alg 3 detects the false data and eliminates them from estimation; False data have influence on how fast the estimates converge

22. Conclusions • 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

23. Thank you! baras@isr.umd.edu 301-405-6606 http://www.isr.umd.edu/~baras Questions? 23