Distributed Fault Detection for Interconnected Second-Order Systems with Applications to Power Networks

Transcription

1 Distributed Fault Detection for Interconnected Second-Order Systems with Applications to Power Networks Iman Shames 1 André H. Teixeira 2 Henrik Sandberg 2 Karl H. Johansson 2 1 The Australian National University and NICTA 2 ACCESS Linnaeus Centre, Kungliga Tekniska högskolan (KTH) VIKING April 12, 2010 Shames et al. (ANU,KTH) SCS April 12, / 14

2 Motivation Consensus Protocols in Practice The main objective of such protocol is to achieve an agreement on a certain quantity of interest Example of applications: Formation Control Deployment Distributed estimation Shames et al. (ANU,KTH) SCS April 12, / 14

3 Motivation Consensus Protocols in Practice The main objective of such protocol is to achieve an agreement on a certain quantity of interest Example of applications: Formation Control Deployment Distributed estimation However, these systems are prone to fault. Shames et al. (ANU,KTH) SCS April 12, / 14

4 Motivation Consensus Protocols in Practice The main objective of such protocol is to achieve an agreement on a certain quantity of interest Example of applications: Formation Control Deployment Distributed estimation However, these systems are prone to fault. We want to detect and isolate a fault when it occurrs. Shames et al. (ANU,KTH) SCS April 12, / 14

5 Motivation Consensus Protocols in Practice The main objective of such protocol is to achieve an agreement on a certain quantity of interest Example of applications: Formation Control Deployment Distributed estimation However, these systems are prone to fault. We want to detect and isolate a fault when it occurrs. Application in power networks and multiagent systems Shames et al. (ANU,KTH) SCS April 12, / 14

6 Problem Description Shames et al. (ANU,KTH) SCS April 12, / 14

7 Problem Description Shames et al. (ANU,KTH) SCS April 12, / 14

8 Problem Description Shames et al. (ANU,KTH) SCS April 12, / 14

9 Problem Description How to detect and isolate the fault? Shames et al. (ANU,KTH) SCS April 12, / 14

10 Network Models Consider N agents ξ i (t) = ζ i (t) ζ i (t) = u i (t), Protocol 1: u i (t) = d i m i ζ i (t) + j N i w ij m i ( ξj (t) ξ i (t) ) Protocol 2: u i (t) = j N i w ij [( ξj (t) ξ i (t) ) + γ ( ζ j (t) ζ i (t) )] Shames et al. (ANU,KTH) SCS April 12, / 14

11 Network Models Set x(t) = [ξ 1 (t),, ξ N (t), ζ 1 (t),, ζ N (t)] ẋ(t) = Ax(t) Protocol 1: [ ] 0N I A = N ML D M ( ) 1 1 M = diag,, m 1 m N D = diag (d 1,, d N ) Protocol 2: [ 0N I A = N L γl ], y(t) = C i x(t) L is a well-studied algebraic descriptor of a graph; it is called Laplacian. Shames et al. (ANU,KTH) SCS April 12, / 14

12 Network Models Fault at agent k: ξ k (t) = ζ k (t) + f k (t) ẋ(t) = Ax(t) + b k f f k(t) Shames et al. (ANU,KTH) SCS April 12, / 14

13 Solution Sketch Model-based Fault Detection and Isolation Construct a bank of observers at each node to monitor its neighbours. Basic Ideas: Compute an expected output; Compare and evaluate the real and expected outputs. Shames et al. (ANU,KTH) SCS April 12, / 14

14 Sensing Requirements Suppose double integrator dynamics. Shames et al. (ANU,KTH) SCS April 12, / 14

15 Sensing Requirements Suppose double integrator dynamics. For a given bf k (fault distribution vector), it is required to sense (have an appropriate C i ) such that Shames et al. (ANU,KTH) SCS April 12, / 14

16 Sensing Requirements Suppose double integrator dynamics. For a given bf k (fault distribution vector), it is required to sense (have an appropriate C i ) such that ( ) ( ) rank C i bf k = rank bf k = 1 for all Re(s) 0. rank ([ si2n A b k f C i 0Ñi 1 ]) = 2N + 1 Shames et al. (ANU,KTH) SCS April 12, / 14

17 Power Systems Model The active power flow on a distribution grid without losses. Each bus has dynamics given by the swing equation : M i δ i +D i δ i = j N i w ij sin ( δ i δ j ) +Pmi As δ ij = δ i δ j is small, we have sin ( δ i δ j ) δi δ j Global dynamics of the network can be written as ẋ = Ax + BP m It is in form of a consensus algorithm (Protocol 1 earlier!). Shames et al. (ANU,KTH) SCS April 12, / 14

18 Application to Power Systems Distributed Fault Detection Dynamics of the power grid under a fault at bus k { ẋ = Ax + BPm + b j f f j w i = J i x, (1) where b j f is the j th column of B f = B. Similarly as before: Distributed fault detection can be achieved as before, where each bus has a bank of observers, measuring the output of the neighboring buses. Shames et al. (ANU,KTH) SCS April 12, / 14

19 Simulation AREA2 AREA1 <1> <5> <7> <8> (A) (F) G1 600km 500km 60,000 MVA (G) 500km (B) <6> 500km (D) 500km <2> <3> (C) (E) G2 G3 600km 500km 1,300 4,400 MVA MVA <Bus #> (Line) AREA3 <9> <4> G4 70,000 MVA δ Residuals Calculated at Bus 7 Fault Occurance r 6 r 5 r 8 Phase Angles (rad.) Residuals Time (sec.) Time (sec.) Shames et al. (ANU,KTH) SCS April 12, / 14

20 Simulation 4 3 The considered formation in R 2 at time t=0 2 y (m) x (m) 6 Agents Positions x Coordinate 2 Agents Velocities x Coordinate 5 Fault Occurance 1.5 ζ 3 (t) x (m) 3 m/s ξ 3 (t) Fault Occurance Time (sec.) Time (sec.) Shames et al. (ANU,KTH) SCS April 12, / 14

21 Simulation Agents Positions x Coordinate Detection Time Agents Velocities x Coordinate Detection Time x (m) 5 m/s Time (sec.) Time (sec.) Shames et al. (ANU,KTH) SCS April 12, / 14

22 Concluding Remarks and Future Steps Concluding Remarks: Existence of observers for two major consensus algorithms for double integrator agents. Future Steps: Shames et al. (ANU,KTH) SCS April 12, / 14

23 Concluding Remarks and Future Steps Concluding Remarks: Existence of observers for two major consensus algorithms for double integrator agents. Stability of a position consensus algorithm in a system of inter connected heterogeneous double integrators. Future Steps: Shames et al. (ANU,KTH) SCS April 12, / 14

24 Concluding Remarks and Future Steps Concluding Remarks: Existence of observers for two major consensus algorithms for double integrator agents. Stability of a position consensus algorithm in a system of inter connected heterogeneous double integrators. Having full position or 1 velocity feedback from neighbours, we always can construct an observer at each of the nodes. Future Steps: 1 logical or Shames et al. (ANU,KTH) SCS April 12, / 14

25 Concluding Remarks and Future Steps Concluding Remarks: Existence of observers for two major consensus algorithms for double integrator agents. Stability of a position consensus algorithm in a system of inter connected heterogeneous double integrators. Having full position or 1 velocity feedback from neighbours, we always can construct an observer at each of the nodes. Future Steps: 1 logical or Shames et al. (ANU,KTH) SCS April 12, / 14

26 Concluding Remarks and Future Steps Concluding Remarks: Existence of observers for two major consensus algorithms for double integrator agents. Stability of a position consensus algorithm in a system of inter connected heterogeneous double integrators. Having full position or 1 velocity feedback from neighbours, we always can construct an observer at each of the nodes. Future Steps: How to reduce of states at each observer? 1 logical or Shames et al. (ANU,KTH) SCS April 12, / 14

27 Concluding Remarks and Future Steps Concluding Remarks: Existence of observers for two major consensus algorithms for double integrator agents. Stability of a position consensus algorithm in a system of inter connected heterogeneous double integrators. Having full position or 1 velocity feedback from neighbours, we always can construct an observer at each of the nodes. Future Steps: How to reduce of states at each observer? Classification of observable components of a network. 1 logical or Shames et al. (ANU,KTH) SCS April 12, / 14

28 Concluding Remarks and Future Steps Concluding Remarks: Existence of observers for two major consensus algorithms for double integrator agents. Stability of a position consensus algorithm in a system of inter connected heterogeneous double integrators. Having full position or 1 velocity feedback from neighbours, we always can construct an observer at each of the nodes. Future Steps: How to reduce of states at each observer? Classification of observable components of a network. 1 logical or -Thanks The End Questions? Shames et al. (ANU,KTH) SCS April 12, / 14