Distributed Estimation and Detection for Smart Grid

Transcription

1 Distributed Estimation and Detection for Smart Grid Texas A&M University Joint Wor with: S. Kar (CMU), R. Tandon (Princeton), H. V. Poor (Princeton), and J. M. F. Moura (CMU) 1

2 Distributed Estimation/Detection in Smart Grid Estimation: Large-scale power system state Detection: Attac or anomaly in the system 2

3 ECE, Texas A&M University Distributed Estimation over Linear Dynamical Systems Dynamical field Sensor Networ Field/Signal Process: {x } R M Local observation at sensor n: {y n } Rm x (+1) = Fx + w y n = C n x + v n Applications: Smart Grid, Social Networ, etc... 3

4 Assumptions on the Signal Model We impose the following global assumptions on the signal/observation model: Stabilizability-Assumption (S.1): The pair (F,Q 1/2 ) is stabilizable. The non-degeneracy (positive definiteness) of Q ensures this. Global Detectability-Assumption (D.1): The pair (C, F) is detectable, where C = [C T 1 C T N ]T. Remar 1. We do not assume local detectability at each agent. These assumptions are required even by the centralized estimator to achieve a stable estimation error. We later show that under rather wea conditions on the inter-sensor communication, the global detectability assumption is also sufficient for our distributed scheme to achieve stable estimation errors at each sensor. 4

5 Distributed Filtering Architecture The agents (sensors) need to collaborate for successful estimation of {x }. Basic Sensing and Communication Architecture: Inter-agent communication rate is measured as the lin activation rate Inter-agent communication is assumed perfect Communication is constrained by the networ topology; and only γ > 0 rounds of inter-agent message passing may be allowed per epoch (,( +1) ] on an average. 5

6 M-GIKF Scheme The overall communication rate γ is split over two phases: +1 Comm Sample Slotted Time Estimate swap + Observation Agg Estimate Swapping: Agents perform pairwise swapping of previous epoch s estimates (if the corresponding lin is active) Estimate Swapping guarantees asymptotic stability of local estimation errors under wea global detectability assumptions Observation Aggregation: Agents use the remaining communication to disseminate their instantaneous observations to the neighbors Observation Aggregation improves estimation performance 6

7 Communication Rate Constraint Define: M e (): Numberofsensorcommunicationsatthe-th epochforestimate swapping M o (): Number of sensor communications at the -th epoch for observation aggregation Communication Constraint: Since γ > 0 is given, we require: lim sup 1 1 (M e ()+M o ()) γ a.s. j=0 7

8 GS Protocol: Estimate Swapping Connectivity- Assumption(C.1): The maximal graph(v, E) is connected, where V denotes the set of N agents and E the set of allowable lins. Proposition 1. Under (C.1) we have the following: The sequence {A e ()} of estimate swapping adjacency matrices is independent and identically drawn from a special distribution D The mean adjacency matrix A e is irreducible and aperiodic The average number of inter-sensor communications due to estimate swapping satisfies almost surely (a.s.) M e 1 = lim (1/) i=0 M e () < γ/2 8

9 GS Protocol: Observation Aggregation Adjacency matrix sequence drawn following a Poisson process of rate γ/2 Proposition 2. Under (C.1) we have the following: The average number of inter-agent communications due to observation aggregation per epoch is γ/2. The sequence {I n } denotes the set of observations (w.r.t. node indices) available at sensor n at the end of the epochs; then for every epoch and every sensor n, P(I n = [1,,N]) > 0 9

10 Reasonable Communication Scheme Definition 1. A communication scheme (for estimate swapping and observation aggregation) is said to be reasonable if (E.1) The estimate swapping adjacency matrices {A e ()} are i.i.d. The mean matrix A is doubly stochastic, irreducible, and aperiodic. (E.2) The sequences {I n } are i.i.d. for each n, independent of the estimate swapping and satisfy P(I n = [1,,N]) > 0. (E.3) The average number of inter-sensor communications (including both the estimate swapping and observation aggregation steps) per epoch is less than or equal to γ, where γ > 0 is a predefined upper bound on the communication rate. Remar 2. The previous GS protocol is a reasonable scheme under (C.1). 10

11 M-GIKF: Estimate Update Define: ( x n 1, P n): Estimate (state) at sensor n of x based on information till time 1 n : Neighbor of sensor n at time w.r.t. Ae (). With estimate swapping: swap ( x n 1, P n ) and With observation aggregation: sensor n collects: y I n Update Rule: x n +1 = E [ ] x +1 x n 1, P n,y I n ( x n 1, P n ( )( T P +1 n = E[ x +1 x n +1 x +1 x +1 ) n x n 1, P n,y I n ) ; ] 11

12 M-GIKF: Estimate Update (Contd.) The filtering steps can be implemented through time-varying Kalman filter recursions The sequence { P n } of conditional predicted error covariance matrices at sensor n satisfies the random Riccati recursion: ( ) P +1 n = F P n F T +Q F P n Cn T C n P n Cn T +R n 1Cn P n F T The sequence { P n } is random due to the random neighborhood selection functions n and I n The goal is to study asymptotic properties of { P n In what sense { P n } is stable In what sense they reach agreement } at every sensor n 12

13 Switched Riccati Iterates Let P denote a generic subset of [1,,N]. Define: C P : The stac of C j s for all j P f P ( ): The Riccati operator given by f P (X) = FXF T +Q FXC T P ( CP XC T P +R n ) 1CP XF T The conditional prediction error covariances are then updated as: ) P n ( +1) = f I n ( P n (), n Remar 3. The sequence { P n ()} evolves as a random Riccati iterate with non-stationary switching. 13

14 M-GIKF: Main Results on Convergence Theorem 1. Consider the M-GIKF under assumptions(s.1), (D.1), (C.1), and (E.1)-(E.3). Then, (1) Let q be a uniformly distributed random variable on [1,,N] independent of the sequences {A e ()} and {I n}. Then, the sequence { Pq ()} converges wealy to an invariant distribution µ γ on S N +, i.e., P q () = µ γ. (2) The performance approaches the centralized one exponentially over γ. (3) For each n, the sequence of conditional error covariances { P n ()} stochastically bounded, is lim sup P( P n () J) = 0. J T + 14

15 Proof Methodology An appropriate stationary modification of the switched Riccati iterate governing the covariance evolution leads to a hypothetical Random Dynamical System (RDS) in the sense of Arnold. The RDS thus constructed is shown to be order preserving and strongly sublinear. Together with the global detectability and networ connectivity assumptions, the wea convergence of the stationary hypothetic RDS is established. The ergodicity of the actual covariance sequence { P n ()} is then applied to obtain the same convergence results for the original non-stationary system. 15

16 Summary of Estimation Results Stability of distributed Kalman filtering errors is established under assumptions of networ connectivity and global detectability. The analysis requires a new approach to the random Riccati equation (with non-stationary switching). Distributional convergence of the random Riccati equation is established; Approaching centralized performance exponentially fast over the intersensor communication rate. 16

17 Distributed Detection N sensors decide between two hypotheses: H 0 vs. H 1. Sensor n observes Y n () at time ; observation vector: Y() = [Y 1 (),...,Y N ()] R N Conditional PDFs f 0 and f 1 : Under H l : f l (Y())dν, l = 0,1. Global Kullbac-Lieber (KL) divergence D(P 0 P 1 ) vs. D(P 1 P 0 ) : ( )] ( ) f0 (Y()) f0 (Y) D(P 0 P 1 ) = E 0 [ln = ln f 0 (Y)dν(Y). f 1 (Y()) R N f 1 (Y) 17

18 Mixed Time Scale Distributed Detection: Algorithm MD Each sensor maintains a local decision variable: X n (). Connected sensor pair (i,j) communicates as U i,j () = X i ()+W i,j (). Local decision: H 1 if X n () > 0; otherwise H 0. Type-I error: P 1 e(n,) = P 0 (X n () > 0), and the Type-II error: P 2 e(n,) = P 1 (X n () 0). 18

19 Local Decision Variable X n () At time, local observation ) generates local intelligence (LLR): S n () = log. ( f1 (Y n ()) f 0 (Y n ()) At time +1, local decision variable is updated (mixed time scale) as X n ( + 1) = X n () β() j Ωn = X n () β() = j Ωn j Ωn [ Xn () U n,j () ] + α() [ S n () X n () ] [X n () X j () W n,j ()] +α() [ S n () X n () ] [1 β() 1 α() ]X n () + β() X j () +β() j Ωn W n,j () + α()s n (). j Ωn 19

20 Key Assumptions (E.1): The sensor observations are conditionally independent, i.e., N Under H l : f l (Y())dν = f n,l (Y n ())dν, l = 0,1, (E.2): We assume global detectability, i.e., D(P 0 P 1 ) > 0 (and D(P 1 P 0 ) > 0). n=1 (E.3): The inter-sensor communication networ is connected. (E.4): The time varying weight sequences {β()} =0 and {α()} =0 associated with the update process satisfy β() = b 0 ( +1) τ, 1 2 < τ < 1, and α() = a ( +1). (lim β()/α() = : Consensus potential vs. Innovation potential.) 20

21 Main Theorem and Summary Theorem 2. Let assumptions (E.1)-(E.4) hold. Then, lim P1 e(n,) = 0 and lim P2 e(n,) = 0, n = 1,...,N. When local observations are i.i.d., a single time scale distributed algorithm is achievable; With Gaussian noise assumptions, error decay exponent is obtainable. 21

22 About My School and Research Group Texas A&M University: One of the largest public schools in USA; Dwight Loo Engineering School (USNews Ran 13th) Dept. of Electrical and Computer Engineering (USNews Ran 18th) My research Group: 2 PostDoc, 9 Ph.D students, and 2 MS students 8 NSF and DoD research projects Main topics: networ information theory, cognitive radio networs, and stochastic signal processing for smart grid Openings: Postdocs and students for the coming year 22