distributed fault detection for interconnected second ... · solution sketch model-based fault...

Distributed Fault Detectionfor Interconnected Second-Order Systems

with Applications to Power Networks

Iman Shames1

André H. Teixeira2 Henrik Sandberg2 Karl H. Johansson2

1The Australian National University and NICTA

2ACCESS Linnaeus Centre, Kungliga Tekniska högskolan (KTH)

VIKING

April 12, 2010Shames et al. (ANU,KTH) SCS April 12, 2010 1 / 14

MotivationConsensus Protocols in Practice

The main objective of such protocol is toachieve an agreement on a certainquantity of interestExample of applications:

I Formation ControlI DeploymentI Distributed estimation

However, these systems are prone to fault.We want to detect and isolate a fault whenit occurrs.Application in power networks andmultiagent systems

Shames et al. (ANU,KTH) SCS April 12, 2010 2 / 14

MotivationConsensus Protocols in Practice

The main objective of such protocol is toachieve an agreement on a certainquantity of interestExample of applications:

I Formation ControlI DeploymentI Distributed estimation

However, these systems are prone to fault.

We want to detect and isolate a fault whenit occurrs.Application in power networks andmultiagent systems

Shames et al. (ANU,KTH) SCS April 12, 2010 2 / 14

MotivationConsensus Protocols in Practice

The main objective of such protocol is toachieve an agreement on a certainquantity of interestExample of applications:

I Formation ControlI DeploymentI Distributed estimation

However, these systems are prone to fault.We want to detect and isolate a fault whenit occurrs.

Application in power networks andmultiagent systems

Shames et al. (ANU,KTH) SCS April 12, 2010 2 / 14

MotivationConsensus Protocols in Practice

The main objective of such protocol is toachieve an agreement on a certainquantity of interestExample of applications:

I Formation ControlI DeploymentI Distributed estimation

However, these systems are prone to fault.We want to detect and isolate a fault whenit occurrs.Application in power networks andmultiagent systems

Shames et al. (ANU,KTH) SCS April 12, 2010 2 / 14

Problem Description

How to detect and isolate the fault?

Shames et al. (ANU,KTH) SCS April 12, 2010 3 / 14

Problem Description

How to detect and isolate the fault?

Shames et al. (ANU,KTH) SCS April 12, 2010 3 / 14

Problem Description

How to detect and isolate the fault?

Shames et al. (ANU,KTH) SCS April 12, 2010 3 / 14

Problem Description

How to detect and isolate the fault?

Shames et al. (ANU,KTH) SCS April 12, 2010 3 / 14

Network Models

Consider N agents

ξi(t) = ζi(t)

ζi(t) = ui(t),

Protocol 1:ui(t) = − di

miζi(t) +

∑j∈Ni

wij

mi

(ξj(t)− ξi(t)

)Protocol 2:

ui(t) =∑j∈Ni

wij[(ξj(t)− ξi(t)

)+ γ

(ζj(t)− ζi(t)

)]

Shames et al. (ANU,KTH) SCS April 12, 2010 4 / 14

Network ModelsSet x(t) = [ξ1(t), · · · , ξN(t), ζ1(t), · · · , ζN(t)]>

x(t) = Ax(t)

Protocol 1:

A =

[0N IN−ML −DM

]M = diag

(1

m1, · · · , 1

mN

)D = diag (d1, · · · ,dN)

Protocol 2:

A =

[0N IN−L −γL

],

y(t) = Cix(t)

L is a well-studied algebraic descriptor of a graph; it is calledLaplacian.

Shames et al. (ANU,KTH) SCS April 12, 2010 5 / 14

Network Models

Fault at agent k :

ξk (t) = ζk (t) + fk (t)

x(t) = Ax(t) + bkf fk (t)

Shames et al. (ANU,KTH) SCS April 12, 2010 6 / 14

Solution SketchModel-based Fault Detection and Isolation

Construct a bank of observers at each node to monitor itsneighbours.

Basic Ideas:I Compute an expected output;I Compare and evaluate the real and expected outputs.

Shames et al. (ANU,KTH) SCS April 12, 2010 7 / 14

Sensing Requirements

Suppose double integrator dynamics.

For a given bkf (fault distribution vector), it is required to sense

(have an “appropriate” Ci ) such that

rank(

Cibkf

)= rank

(bk

f

)= 1

rank([

sI2N − A bkf

Ci 0Ni×1

])= 2N + 1

for all Re(s) ≥ 0.

Shames et al. (ANU,KTH) SCS April 12, 2010 8 / 14

Sensing Requirements

Suppose double integrator dynamics.For a given bk

f (fault distribution vector), it is required to sense(have an “appropriate” Ci ) such that

rank(

Cibkf

)= rank

(bk

f

)= 1

rank([

sI2N − A bkf

Ci 0Ni×1

])= 2N + 1

for all Re(s) ≥ 0.

Shames et al. (ANU,KTH) SCS April 12, 2010 8 / 14

Sensing Requirements

Suppose double integrator dynamics.For a given bk

f (fault distribution vector), it is required to sense(have an “appropriate” Ci ) such that

rank(

Cibkf

)= rank

(bk

f

)= 1

rank([

sI2N − A bkf

Ci 0Ni×1

])= 2N + 1

for all Re(s) ≥ 0.

Shames et al. (ANU,KTH) SCS April 12, 2010 8 / 14

Power Systems Model

The active power flow on adistribution grid without losses.Each bus has dynamics givenby the ”swing equation“:

Mi δi+Di δi = −∑j∈Ni

wij sin(δi − δj

)+Pmi

As δij = δi − δj is small, wehave sin

(δi − δj

)≈ δi − δj Global dynamics of the

network can be written as

x = Ax + BPmIt is in form of a consensus algorithm (Protocol 1 earlier!).

Shames et al. (ANU,KTH) SCS April 12, 2010 9 / 14

Application to Power SystemsDistributed Fault Detection

Dynamics of the power grid under a fault at bus k{x = Ax + BPm + bj

f fjwi = Jix,

(1)

where bjf is the j th column of Bf = B.

Similarly as before:Distributed fault detection can be achieved as before, where eachbus has a bank of observers, measuring the output of theneighboring buses.

Shames et al. (ANU,KTH) SCS April 12, 2010 10 / 14

Simulation<Bus #>(Line)

G1

AREA2

<1> <5>

<6>

500k

m

60,000MVA

AREA1

<4>

G4

AREA3<9>

70,000MVA

G2

<7> <8>

G3

(F)(A)

600km 500km

(G)

500km(B)

(D)

500km

<2> <3>(C) (E)

600km 500km1,300MVA

4,400MVA

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

Phase A

ngle

s (

rad.)

Time (sec.)

δ6

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (sec.)

Resid

uals

Residuals Calculated at Bus 7

r6

r5

r8

FaultOccurance

Shames et al. (ANU,KTH) SCS April 12, 2010 11 / 14

Simulation

0 0.5 1 1.5−2

−1

0

1

2

3

4

x (m)

y (m

)

The considered formation in R2 at time t=0

1

3

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6Agents Positions x Coordinate

Time (sec.)

x (

m)

FaultOccurance

ξ3(t)

0 1 2 3 4 5 6 7 8 9 10−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (sec.)

m/s

Agents Velocities x Coordinate

FaultOccurance

ζ3(t)

Shames et al. (ANU,KTH) SCS April 12, 2010 12 / 14

Simulation

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10Agents Positions x Coordinate

Time (sec.)

x (

m)

DetectionTime

0 1 2 3 4 5 6 7 8 9 10 110

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Agents Velocities x Coordinate

m/s

Time (sec.)

DetectionTime

Shames et al. (ANU,KTH) SCS April 12, 2010 13 / 14

Concluding Remarks and Future Steps

Concluding Remarks:Existence of observers for two major consensus algorithms fordouble integrator agents.

Stability of a position consensus algorithm in a system of interconnected heterogeneous double integrators.Having full position or1 velocity feedback from neighbours, wealways 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.

-Thanks–The End

—Questions?

1logical or

Shames et al. (ANU,KTH) SCS April 12, 2010 14 / 14

Concluding Remarks and Future Steps

Concluding Remarks:Existence of observers for two major consensus algorithms fordouble integrator agents.Stability of a position consensus algorithm in a system of interconnected heterogeneous double integrators.

Having full position or1 velocity feedback from neighbours, wealways 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.

-Thanks–The End

—Questions?

1logical or

Shames et al. (ANU,KTH) SCS April 12, 2010 14 / 14

Concluding Remarks and Future Steps

Concluding Remarks:Existence of observers for two major consensus algorithms fordouble integrator agents.Stability of a position consensus algorithm in a system of interconnected heterogeneous double integrators.Having full position or1 velocity feedback from neighbours, wealways 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.

-Thanks–The End

—Questions?

1logical orShames et al. (ANU,KTH) SCS April 12, 2010 14 / 14

Concluding Remarks and Future Steps

Concluding Remarks:Existence of observers for two major consensus algorithms fordouble integrator agents.Stability of a position consensus algorithm in a system of interconnected heterogeneous double integrators.Having full position or1 velocity feedback from neighbours, wealways 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.

-Thanks–The End

—Questions?

1logical orShames et al. (ANU,KTH) SCS April 12, 2010 14 / 14

Concluding Remarks and Future Steps

Concluding Remarks:Existence of observers for two major consensus algorithms fordouble integrator agents.Stability of a position consensus algorithm in a system of interconnected heterogeneous double integrators.Having full position or1 velocity feedback from neighbours, wealways 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.

-Thanks–The End

—Questions?

1logical orShames et al. (ANU,KTH) SCS April 12, 2010 14 / 14

Concluding Remarks and Future Steps

Concluding Remarks:Existence of observers for two major consensus algorithms fordouble integrator agents.Stability of a position consensus algorithm in a system of interconnected heterogeneous double integrators.Having full position or1 velocity feedback from neighbours, wealways 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.

-Thanks–The End

—Questions?

1logical orShames et al. (ANU,KTH) SCS April 12, 2010 14 / 14

Concluding Remarks and Future Steps

Concluding Remarks:Existence of observers for two major consensus algorithms fordouble integrator agents.Stability of a position consensus algorithm in a system of interconnected heterogeneous double integrators.Having full position or1 velocity feedback from neighbours, wealways 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.

-Thanks–The End

—Questions?

1logical orShames et al. (ANU,KTH) SCS April 12, 2010 14 / 14