Multi-Sensor Networked Estimation in Electric Power Grids
Multi-Sensor Networked Estimation in Electric Power Grids Bei Yan and Hanoch Lev-Ari Department of Electrical and Computer Engineering Northeastern University
Lower Bound
Problem Formulation A continuous-time discrete-measurement linear dynamic system: ˙ x(t) = Ax(t) + w (t) y(tn) = C(tn)x(tn) + η(tn) Measurement update (at time tn): ˆ+(tn) = x ˆ−(tn) + P−(tn)C ∗(tn) x ˆ−(tn)) (C(tn)P−(tn)C ∗(tn) + R(tn))−1(y(tn) − C(tn)x P+(tn) = P−(tn) − P−(tn)C ∗(tn) ∗ −1 (C(tn)P−(tn)C (tn) + R(tn)) C(tn)P−(tn) I
Time update (in the interval tn ≤ t < tn+1): A(t−tn ) ˆ ˆ x (t) = e x+(tn) Z t−tn A(t−tn ) A∗(t−tn ) Aτ A∗τ P(t) = e P+(tn)e + e Qe dτ
s
and obtain a necessary condition for the convergence of EP−(tn) σmax Re(λi ) < (8) 2 In particular, in the multi-sensor Bernoulli case with packet drop probability pk for sensor k , we have ( p0 i =0 Pr(Ts (n) = i∆) = (1 − p0)(1 − p)p i−1 i ≥ 1 where (for L sensors) p =
(1)
(2)
(7)
L Q
L Q
pk and p0 =
k =1
(1−pk )
k =1 L P
.
(1−pk )
The characteristic function of the superposed MSTP becomes (1 − p)e s∆ φTs (s) = p0 + (1 − p0) 1 − pe s∆ and its region of convergence is Re(s) < σmax , where 1 σmax = − ∆ ln p . The necessary stability condition (8) becomes
(3)
p =
L Y
8000
(5)
pk < e
−2∆ max Re(λi ) i
720
700
680 6000
5000
4000
660
640
2000
, pc
(9)
600 1000
0
580 0
1
2
3
4
5
6
7
8
9
10
1.5
1.52
Average sampling rate F
1.54
1.56
1.58
1.6
1.62
Average sampling rate F
av
av
Figure: P−,av vs Fav in two-sensor Bernoulli-drop case: two unstable eigenvalues in A
Upper Bound
∗ ∗ XCk (Ck XCk
(10) −1
fk (X ) = X − + Rk ) Ck X (11) where “k ” denotes the sensor associated with tn. Since fk (X ) is a down-convex, monotone increasing function, so −1 −∗ ∗ ¯ gk (X ) = T {ΩTs ◦ [T fk (X )T ]}T + S (12)
and applying ensemble averaging we get the relation
However, for other sets of system parameter values both bounds remain close for all Fav > Fc , as shown in Fig. 2 for 1.25 1 0 20 20 20 A = 0 −0.9 7 , Q = 20 20 20 , 0 0 −0.6 20 20 20 C1 = 1 1 , C2 = 2 3 , R = 2.5 Notice that in this case A has only one unstable eigenvalues. 10000
] (6)
inherits the properties. We can combine (10) with (6) to get N X ¯k E{gk (P−(tn))} F EP−(tn+1) = E{gk (P−(tn))} =
where A = T ΛT and Λ = diag{λi } is a diagonal matrix constructed from the eigenvalues of A. Here ◦ denotes the Schur-Hadamard product and λ∗m)]M l,m=1
ΩTs = [φTs (λl + 1 ∗ M (φ (λ + λ ) − 1)] 0Ts = [ T l s m l,m=1 ∗ λ l + λm where φTs (s) = E{e sTs(n)} is the characteristic function of the (stationary) sampling interval Ts (n).
8000
≤
N X
¯k gk (EP−(tn)) , G(EP−(tn)) F
650
600
(13)
k =1
PN
ξ=0.7 ξ=0.6 ξ=0.52 ξ=0.28 ξ=0.21
700
7000
k =1
−1
ξ=0.7 ξ=0.6 ξ=0.52 ξ=0.28 ξ=0.21
9000
trace(P−,av)
= ΩTs ◦ [T EP+(tn−1)T −1 −∗ +0Ts ◦ [T QT ]
ξ=0.7 ξ=0.6 ξ=0.52 ξ=0.28 ξ=0.21
7000
(4)
0
EP−(tn)T
ξ=0.7 ξ=0.6 ξ=0.52 ξ=0.28 ξ=0.21
9000
k =1
At the measurement instants {tn}, we replace (5) by Z Ts(n) ATs (n) A∗Ts (n) Aτ A∗τ P−(tn) = e P+(tn−1)e + e Qe dτ T
=
620
P+(tn) = fk (P−(tn))
−∗
The system parameters are 1.25 0 20 20 A= ,Q = , C1 = 1 1 , C2 = 2 3 , R = 2.5 1 1.1 20 20 Notice that we have two unstable eigenvalues of A, and the ¯ system is both controllable and observable. The lower bound tr(S) increases dramatically as Fav approaches the critical value, i.e. 1−p ln p ∆c = −2 max Re(λi ) and Fc = (1−p0)∆c (≈ 1.25), and diverge for Fav ≤ Fc . On the other hand, the upper bound diverges for a bigger value of Fav (≈ 2.4). ξ=
3000
Dynamics of Average Error Covariance
−1
p+p0 . 1−p0
10000
We now turn our attention to the measurement update (3) as
−∗
VarTs (n) Tav 2
k =1
0
−1
We use here the Bernoulli MSTP to illustrate our results (Fig. 1). ¯ and V ¯ = limn→∞ Vn , as a function of the We plot the trace of S 1−p average sampling rate Fav = (1−p0)∆ with different relative variance
¯k gk (X ) and F ¯k , PNFk , where Fk is the where G(X ) , k =1 F i=1 Fi average sampling rate of the sensor “k ”. The inequality (13) implies that EP−(tn) ≤ Vn where Vn = G(Vn−1). In contrast to the lower bound which holds for any MSTP, the upper bound EP−(tn) ≤ Vn holds when {Ts (n)} are i.i.d. This property need not hold for every superposition of renewal processes. Fortunately, the multi-sensor Bernoulli drop pattern has i.i.d property.
6000
trace(P−,av)
I
¯ for all n, where We deduce from (6) that EP−(tn) ≥ S −1 −∗ ∗ ¯ S = T [0T ◦ (T QT )]T
trace(P−,av)
In Electric Power Systems, distributed control and sensor network applications have been widely used. When data is sent via unreliable communication channels, the effect of communication delays and loss of information cannot be neglected. In this poster, we focus on the effects of irregular measurement timing patterns on the performance of a continuous-discrete Kalman filter estimator. We implement Bernoulli drop multi-sensor timing pattern (MSTP) and analyze the convergence of the average error covariance.
Simulation Results
trace(P−,av)
Abstract
5000
4000
550
500 3000
2000 450 1000
0
400 0
1
2
3
4
5
6
7
Average sampling rate F
av
8
9
10
1.5
1.55
1.6
1.65
1.7
1.75
Average sampling rate F
av
Figure: P−,av vs Fav in two-sensor Bernoulli-drop case: one unstable eigenvalue in A
We can draw the conclusion that the steady-state average error covariance depends primarily on the ensemble-averaged sampling interval with a minor dependence on the variance.
Lower Bound
Problem Formulation A continuous-time discrete-measurement linear dynamic system: ˙ x(t) = Ax(t) + w (t) y(tn) = C(tn)x(tn) + η(tn) Measurement update (at time tn): ˆ+(tn) = x ˆ−(tn) + P−(tn)C ∗(tn) x ˆ−(tn)) (C(tn)P−(tn)C ∗(tn) + R(tn))−1(y(tn) − C(tn)x P+(tn) = P−(tn) − P−(tn)C ∗(tn) ∗ −1 (C(tn)P−(tn)C (tn) + R(tn)) C(tn)P−(tn) I
Time update (in the interval tn ≤ t < tn+1): A(t−tn ) ˆ ˆ x (t) = e x+(tn) Z t−tn A(t−tn ) A∗(t−tn ) Aτ A∗τ P(t) = e P+(tn)e + e Qe dτ
s
and obtain a necessary condition for the convergence of EP−(tn) σmax Re(λi ) < (8) 2 In particular, in the multi-sensor Bernoulli case with packet drop probability pk for sensor k , we have ( p0 i =0 Pr(Ts (n) = i∆) = (1 − p0)(1 − p)p i−1 i ≥ 1 where (for L sensors) p =
(1)
(2)
(7)
L Q
L Q
pk and p0 =
k =1
(1−pk )
k =1 L P
.
(1−pk )
The characteristic function of the superposed MSTP becomes (1 − p)e s∆ φTs (s) = p0 + (1 − p0) 1 − pe s∆ and its region of convergence is Re(s) < σmax , where 1 σmax = − ∆ ln p . The necessary stability condition (8) becomes
(3)
p =
L Y
8000
(5)
pk < e
−2∆ max Re(λi ) i
720
700
680 6000
5000
4000
660
640
2000
, pc
(9)
600 1000
0
580 0
1
2
3
4
5
6
7
8
9
10
1.5
1.52
Average sampling rate F
1.54
1.56
1.58
1.6
1.62
Average sampling rate F
av
av
Figure: P−,av vs Fav in two-sensor Bernoulli-drop case: two unstable eigenvalues in A
Upper Bound
∗ ∗ XCk (Ck XCk
(10) −1
fk (X ) = X − + Rk ) Ck X (11) where “k ” denotes the sensor associated with tn. Since fk (X ) is a down-convex, monotone increasing function, so −1 −∗ ∗ ¯ gk (X ) = T {ΩTs ◦ [T fk (X )T ]}T + S (12)
and applying ensemble averaging we get the relation
However, for other sets of system parameter values both bounds remain close for all Fav > Fc , as shown in Fig. 2 for 1.25 1 0 20 20 20 A = 0 −0.9 7 , Q = 20 20 20 , 0 0 −0.6 20 20 20 C1 = 1 1 , C2 = 2 3 , R = 2.5 Notice that in this case A has only one unstable eigenvalues. 10000
] (6)
inherits the properties. We can combine (10) with (6) to get N X ¯k E{gk (P−(tn))} F EP−(tn+1) = E{gk (P−(tn))} =
where A = T ΛT and Λ = diag{λi } is a diagonal matrix constructed from the eigenvalues of A. Here ◦ denotes the Schur-Hadamard product and λ∗m)]M l,m=1
ΩTs = [φTs (λl + 1 ∗ M (φ (λ + λ ) − 1)] 0Ts = [ T l s m l,m=1 ∗ λ l + λm where φTs (s) = E{e sTs(n)} is the characteristic function of the (stationary) sampling interval Ts (n).
8000
≤
N X
¯k gk (EP−(tn)) , G(EP−(tn)) F
650
600
(13)
k =1
PN
ξ=0.7 ξ=0.6 ξ=0.52 ξ=0.28 ξ=0.21
700
7000
k =1
−1
ξ=0.7 ξ=0.6 ξ=0.52 ξ=0.28 ξ=0.21
9000
trace(P−,av)
= ΩTs ◦ [T EP+(tn−1)T −1 −∗ +0Ts ◦ [T QT ]
ξ=0.7 ξ=0.6 ξ=0.52 ξ=0.28 ξ=0.21
7000
(4)
0
EP−(tn)T
ξ=0.7 ξ=0.6 ξ=0.52 ξ=0.28 ξ=0.21
9000
k =1
At the measurement instants {tn}, we replace (5) by Z Ts(n) ATs (n) A∗Ts (n) Aτ A∗τ P−(tn) = e P+(tn−1)e + e Qe dτ T
=
620
P+(tn) = fk (P−(tn))
−∗
The system parameters are 1.25 0 20 20 A= ,Q = , C1 = 1 1 , C2 = 2 3 , R = 2.5 1 1.1 20 20 Notice that we have two unstable eigenvalues of A, and the ¯ system is both controllable and observable. The lower bound tr(S) increases dramatically as Fav approaches the critical value, i.e. 1−p ln p ∆c = −2 max Re(λi ) and Fc = (1−p0)∆c (≈ 1.25), and diverge for Fav ≤ Fc . On the other hand, the upper bound diverges for a bigger value of Fav (≈ 2.4). ξ=
3000
Dynamics of Average Error Covariance
−1
p+p0 . 1−p0
10000
We now turn our attention to the measurement update (3) as
−∗
VarTs (n) Tav 2
k =1
0
−1
We use here the Bernoulli MSTP to illustrate our results (Fig. 1). ¯ and V ¯ = limn→∞ Vn , as a function of the We plot the trace of S 1−p average sampling rate Fav = (1−p0)∆ with different relative variance
¯k gk (X ) and F ¯k , PNFk , where Fk is the where G(X ) , k =1 F i=1 Fi average sampling rate of the sensor “k ”. The inequality (13) implies that EP−(tn) ≤ Vn where Vn = G(Vn−1). In contrast to the lower bound which holds for any MSTP, the upper bound EP−(tn) ≤ Vn holds when {Ts (n)} are i.i.d. This property need not hold for every superposition of renewal processes. Fortunately, the multi-sensor Bernoulli drop pattern has i.i.d property.
6000
trace(P−,av)
I
¯ for all n, where We deduce from (6) that EP−(tn) ≥ S −1 −∗ ∗ ¯ S = T [0T ◦ (T QT )]T
trace(P−,av)
In Electric Power Systems, distributed control and sensor network applications have been widely used. When data is sent via unreliable communication channels, the effect of communication delays and loss of information cannot be neglected. In this poster, we focus on the effects of irregular measurement timing patterns on the performance of a continuous-discrete Kalman filter estimator. We implement Bernoulli drop multi-sensor timing pattern (MSTP) and analyze the convergence of the average error covariance.
Simulation Results
trace(P−,av)
Abstract
5000
4000
550
500 3000
2000 450 1000
0
400 0
1
2
3
4
5
6
7
Average sampling rate F
av
8
9
10
1.5
1.55
1.6
1.65
1.7
1.75
Average sampling rate F
av
Figure: P−,av vs Fav in two-sensor Bernoulli-drop case: one unstable eigenvalue in A
We can draw the conclusion that the steady-state average error covariance depends primarily on the ensemble-averaged sampling interval with a minor dependence on the variance.
