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, 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 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 2

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 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.

5 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 6

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: 7

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 v j (k) is the measurement noise assumed Gaussian with zero mean and covariance matrix R j 8

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 x 0 is assumed to be a Gaussian distribution with mean μ 0 and covariance matrix P 0 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 9

Distributed Estimation To compute an accurate estimate of the state x (k), using: local measurements y j (k); information received from the PMUs in its communication neighborhood; confidence in the information received from other PMUs provided by the trust model PMU GPS Satellite N multiple recording sites (PMUs) to measure the output signals 10

11 Trust Model To each information flow (link) j  i, we attach a positive value T ij, 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 Kalman Filtering 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 Accuracy Dependent Consensus Step We define the trust value T ij in terms of the estimation error given by the standard Kalman filter: Remark: Although M i is not the true covariance of the estimation error, it reflects the observability (through C i ) and accuracy (through R i ) of the PMU i Assumption: (A, C i ) detectable

15 Distributed Estimation with Reliability Dependent Consensus Step We assume some PMUs may send false information due to malfunctions or attacks; Update mechanism for T ij is based on belief divergence (Kerchove, 2007), which shows how far a received estimate is from the other received estimates: where N i is the number of neighbors of PMU i

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

17 Distributed Estimation with Reliability Dependent Consensus Step

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

19 Data from a practical example (Lee and Poon, 1990), which has two modes at ω 1 =0.4Hz and ω 2 = 0.5Hz. The power system model employs five measurements, where each PMU is installed over a line connected to one generator Simulations G1 G5 G2 G3 G4

20 Distributed Kalman Filtering with accuracy dependent consensus step White noise with different SNR was added to each measurement Simulations estimating parameter a 1 estimating parameter σ 1 In Alg 2, larger weight is given to information coming from PMUs with small variance of the estimation error

estimating parameter σ 1 estimating parameter a 1 21 Distributed estimation with reliability dependent consensus 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

22 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

23 Thank you! Questions?