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Measurement-Based Methods for Model Reduction,
Identification, and Distributed Optimization of
Power Systems
Seyed Behzad Nabavi
Department of Electrical and Computer Engineering
North Carolina State University
24/4/2015
1/1
Model Reduction, Identification, and Distributed Optimization of Power Systems
c 2015 by S. Nabavi
Part I– Identification of Dynamic
Reduced-Order Models of Power Systems
2/1
c 2015 by S. Nabavi
Identification of Dynamic Reduced-Order Models of Power Systems
Introduction
Introduction
Mathematical modeling of dynamic
equivalents of large-scale electric power
systems has seen some 40 years of long
and rich research history.
Chow and Kokotovic established the
relationship between the slow coherency
and weak connections using singular
perturbation theory.
Slow coherency arises from the slower
inter-area modes. These interarea modes,
Figure: R. Podemore: Coherency in
if not properly damped, lead to system
Power Systems
separation and extensive loss of load.
3/1
c 2015 by S. Nabavi
Introduction
Model-based Dynamic Equivalencing
Real-Time
Monitoring
Recent
evidences Wide-Area
of blackouts have
shown the discrepancy between the
offline models and the response of the system.
W. Winter, K. Elkington, G. Bareux, and J. Kostevc, “Pushing the Limits: Europe's New Grid: Innovative Tools to Combat Transmission Bottlenecks and Reduced Inertia," Power and Energy Magazine, 13(1), 2015
4/1
c 2015 by S. Nabavi
Introduction
Model-based Dynamic Equivalencing
Dynamic equivalencing has seen 40 years of active research:
–
–
–
–
–
linear modal decomposition [Undrill, 71]
circuit-theoretic approaches [de Mello, 75]
machine aggregation [Germond, 78]
enumerative clustering algorithms [Zaborsky, 82]
software programs such as DYNEQ and DYNRED [Price, 95]
Model based methods:
– need the exact knowledge of the entire power system model,
– are computationally challenging,
– are based on idealistic assumption about system structure and clustering.
5/1
c 2015 by S. Nabavi
Introduction
Measurement-based Dynamic Equivalencing
PMUs provide high-resolution GPS-synchronized three-phase
measurements of voltage, current, phasor, and frequency.
System operators are, therefore, inclining more towards online models
constructed from PMU (Phasor Measurement Unit) measurements.
We next propose two algorithms to identify these dynamic equivalent
models using PMU measurements:
* Identification of the equivalent linear models
* Identification of the equivalent nonlinear DAE models
G8
G10
PMU
8
10
26
G4s
29
28
25
G1s
9
27
31
24
G9
G4s
Ips4
Vps4
18
39
17
G1
32
16
6
PMU
15
14
33
1
34
20
7
30
PMU
PMU
38
3
2
G2
Vps3
Ips3
G3s
23
19
13
11
V s
Ips 1 p1
22
35
36
37
G1s
G6
21
12
4
5
G3
G4
G7
G2s
G3s
G5
6/1
Vps2
I s
p2
G2s
c 2015 by S. Nabavi
Power System Model Reduction
Power System Swing Equation
Nonlinear Electromechanical Model:
δ̇i (t) = ωs (ωi (t) − 1),
Mi ω̇i (t) = Pmi − Pei (t) − Di (ωi (t) − 1),
Pe (t) +
k
X
l∈Nk
Pkl (t) − PL (t) = 0, Qe (t) +
k
k
X
l∈Nk
Qkl (t) − QL (t) = 0
k
Linearized Kron-Reduced Model (around (δi0 , 1)):
"
∆δ̇(t)
∆ω̇(t)
#
"
=
0n×n
ωs In×n
#"
∆δ(t)
M −1 L −M −1 D
∆ω(t)
|
{z
}
A
T
∆δ , ∆δ1 · · · ∆δn ,
#
+ Bd(t),
T
∆ω , ∆ω1 · · · ∆ωn , ∆δ, ∆ω ∈ Rn
M = diag(Mi ) ∈ Rn×n , D = diag(Di ) ∈ Rn×n , d(t) : unknown disturbance
[L]i,j = Ei Ej (Gij cos(δi0 − δj0 ) − Bij sin(δi0 − δj0 )) i 6= j,
[L]i,i = −
n
X
[L]i,k ,
k =1
7/1
c 2015 by S. Nabavi
Linear Dynamic Equivalent Models
G8
8
G10
10
PMU
26
G1s
29
28
9
G9
27
25
31
24
18
39
G1
17
32
16
6
PMU
15
14
33
1
21
22
12
34
23
19
35
36
13
11
37
20
3
G2
Area
Area2
Area
Area
7
30
PMU
PMU
38
2
G6
G4s
G7
4
5
G3
G4
G5
8/1
G2s
Aggregated
Transmission
Network Graph
G3s
c 2015 by S. Nabavi
Identification of Linear Equivalent Models
The reduced-order model:
"
∆δ̇ s (t)
∆ω̇ s (t)
#
"
=
0r ×r
(M s )−1 Ls
|
ωs Ir ×r
#"
∆δ s (t)
#
−(M s )−1 D s
∆ω s (t)
{z
}
As
+ B s d(t).
Assumptions:
– The area partitioning for our system is known apriori.
– There is at least one PMU at a generator bus in each area (S).
– As and B s are a controllable pair.
Objective:
– Finding the equivalent linear model of a power system from yi (t) i ∈ S:
yi (t) = {Ṽi (t), Ĩi,j (t)}, i ∈ S, j ∈ Ni .
Proposed Identification Steps:
– Extract ∆δks (t) for each area k from yi (t), i ∈ S.
G1s
G2s
– Identification of As .
9/1
G4s
Area
Area2
Area
Area
Aggregated
Transmission
Network Graph
G3s
c 2015 by S. Nabavi
Extraction of ∆δks (t) for Each Area k
Step 1: Extract ∆δi (t) from yi (t):
Ei (t)∠δi (t) =
jxd0 i Ii (t)∠φi (t)
jxd i
Ii
Vi i
Ei  i
+ Vi (t)∠θi (t)
⇒ δ̂i (t) = ∠(jxd0 i Ii (t)∠φi (t) + Vi (t)∠θi (t))
∆δ̂i (t) = δ̂i (t) − δ̂i (t0 ).
Step 2: Extract ∆δ̂ks (t) from ∆δ̂i,k (t), (generator i belonging to area k ):
0
∆δi,k (t) = ∆δi,k (t) +
r −1
X
ρil e
(−σl +jΩl )t
∗ (−σl −jΩl )t
+ ρil e
n−1
X
ρil e
(−σl +jΩl )t
∗ (−σl −jΩl )t
,
+ ρil e
l=r
l=1
|
+
{z
∆δ s (t), inter-area or slow modes
i,k
}
|
{z
}
∆δ f (t), intra-area or fast modes
i,k
Use a modal decomposition technique such as Prony to decompose
∆δi,k (t)
s
Form ∆δi,k
(t) by retaining only the modes in [0.1,1] Hz.
10 / 1
c 2015 by S. Nabavi
Extraction of ∆δks (t) for Each Area k
G8
8
G10
10
PMU
26
G1s
29
28
9
G9
27
25
31
24
18
39
G1
17
32
16
6
PMU
15
14
33
1
21
22
12
34
23
19
35
36
13
11
37
20
PMU
G2
G7
4
5
3
Area
Area2
Area
Area
7
30
PMU
38
2
G6
G4s
G3
G4
G5
G2s
Aggregated
Transmission
Network Graph
G3s
s
We truncate ∆δi,k (t) to extract ∆δi,k
(t).
From the coherency assumption
s
s
s
∆δ1,k
(t) ≈ ∆δ2,k
(t) ≈ · · · ≈ ∆δm
(t)
k ,k
s
We set ∆δks (t) = ∆δi,k
(t).
11 / 1
c 2015 by S. Nabavi
Identification of As
The reduced-order model:
"
∆δ̇ s (t)
#
∆ω̇ s (t)
"
=
ωs Ir ×r
0r ×r
(M s )−1 Ls
|
#"
∆δ s (t)
#
−(M s )−1 D s ∆ω s (t)
{z
}
As
+ B s d(t).
Solve the following NLS problem (assuming d(t) is a momentary
perturbation at t = t0 ):
Z
min
s
A
tm
k
t1
s ∆δ s (t, As )
∆δ̂ (t) 2
−
k dt
∆ω s (t, As )
∆ω̂ s (t) 2
where,
s
∆δ s (t, As )
∆δ̂ (t1 )
s
= exp(A (t − t1 ))
,
∆ω s (t, As )
∆ω̂ s (t1 )
∆ω̂ s (t) is calculated from the numerical differentiation of ∆δ̂ s (t)
normalized by ωs .
12 / 1
c 2015 by S. Nabavi
Identifiability Analysis of As
Lemma: [Bellman and Astrom-70] Consider the system
ẋ = Ax + Bu, y = Cx
If the matrix C is full column-rank and the system is controllable, then A
and B can be determined uniquely from input output data.
In our identification problem, we assume (As , B s ) to be a controllable pair,
and C = I2n (full column-rank), thus As is identifiable.
More results on identifiability analysis will be provided in Part III (joint
work with Dr. P. P. Khargonekar).
13 / 1
c 2015 by S. Nabavi
A Case Study– NPCC 48 Machine Model
G34 G35
97
96
10
G33
92
1 G4
8
103
G36
91
11
31 102
110
112 94
89
123
83
116 84
G38
6
105
71
90
G24 68
119
G39 120
114
G47
122
13
118
G40
127
124
21
1
29
35
6
8
9
126
125
128
132
32
2
2
14
133
G44
16
130
G43
34
G27
81
75
7
76
77 74
G28
1
5
13
3
138
4
73
135
G46
12
4
31
39
129
G48
G2
30
G13
136
14
17
15
5
37
38
28
134
27
5
0 40
51
G14
42
69
3
6
16
G45
131
139
G9
41
4
G10
66
10
18
9
44
43
24
20
19
17
G18
45
70
64 63
67
137
15
G41
121
12 55
49
2
117
G8
56 G19
48G
46
G26 72
G25
G6
8
61
G11
47
11
G7
25
14
13
15
26
62 58
53
3 G23
106
113 88
85
87
G5
23
11
59
52
G16
107
111
G15 65
12
115
10
22
7
12
G22
86
G42
G37
G3
99
100
57
54
104
109
93
9
G20
60
G21
G17
95
108
G32
140
5
101 G
98
7
33
36
78
79 80
82
G29
G30
16
G1
17
14 / 1
c 2015 by S. Nabavi
A Case Study- NPCC 48 Machine Model
3
∆δ1COI
∆δ1s model-based
∆δ8COI
10
6
∆δ8s model-based
4
0
−1
(deg)
5
1
(deg)
(deg)
2
0
COI
∆δ17
s
model-based
∆δ17
2
0
−2
−5
−2
−4
−10
−3
0
5
10
15
0
5
Time (sec)
3
∆δ1COI
10
−6
0
15
∆δ1s Case 2
∆δ8COI
10
10
15
Time (sec)
6
COI
∆δ17
∆δ8s Case 2
2
s
∆δ17
Case 2
4
0
−1
(deg)
5
1
(deg)
(deg)
5
Time (sec)
0
2
0
−2
−5
−2
−4
−10
−3
0
5
10
15
0
5
Time (sec)
10
Time (sec)
15
−6
0
5
10
15
Time (sec)
Defining the error:
Ja (k ) =
1
tm − t1
Z
tm
|∆δks ,reduced (t) − ∆δks ,actual (t)|dt.
t1
P
Pk Ja (k ) = 10.2232(deg) for the model-based method, and
k Ja (k ) = 4.6017(deg) for our measurement-based method.
15 / 1
c 2015 by S. Nabavi
Identification of the Equivalent DAE Models
The linear equivalent models are in the Kron’s form.
This model is not a very suitable choice for:
1 Identification of the individual equivalent parameters such as inertia (Mi )
2 Shunt controller design purposes
3 Describing the system behavior for large disturbances (transient stability)
Area 1
Area 4
G4s
`
Ips4
Vps4
G1s
Ips 1
Vps1
Vps3
Ips3
G3s
Vps2
I s
p2
Area 2
G2s
Area 3
δ̇is (t)
s s
Mi ω̇i (t)
δ̇i (t) = ωs (ωi (t) − 1),
Mi ω̇i (t) = Pmi − Pei − Di (ωi (t) − 1),
16 / 1
= ωis (t) − ωs ,
s
− Pesi − Dis (ωis (t) − 1),
= Pm
i
c 2015 by S. Nabavi
Identification of the Equivalent DAE Models
Assumptions:
– The area partitioning for our system is known apriori.
– The boundary buses of all areas are equipped with PMUs (denoted by S).
Objective:
– Finding the equivalent DAE model of a power system from yi (t) i ∈ S:
yi (t) = {Ṽi (t), Ĩi,j (t)}, i ∈ S, j ∈ Ni .
Proposed Identification Steps:
G4s
– Finding the equivalent pilot bus voltages and currents.
– Estimating the equivalent area impedances.
Ips4
Vps4
G1s
V s
Ips 1 p1
Vps3
Ips3
G3s
– Estimating the equivalent generator parameters.
– Estimating the inter-area impedances.
Vps2
I s
p2
G2s
17 / 1
c 2015 by S. Nabavi
Equivalent Pilot Bus Voltage and Current
Step 1: Use yk to calculate Ṽpk (t) and Ĩpk (t)
P
Ĩpk (t) , Ipk (t)∠φpk (t) =
X
Ĩi (t), Ṽpk (t) , Vpk (t)∠θpk (t) =
i∈Bk
i∈Bk
Ṽi (t)Ĩi∗ (t)
Ĩp∗k (t)
PMU
PMU
Coherent
Area k
Step 1
Vpk
Coherent
Area k
PMU
18 / 1
Ipk
Step 2
Vpsk
Coherent
Area k
Ipsk
c 2015 by S. Nabavi
Equivalent Pilot Bus Voltage and Current
Step 2: Construction of Ṽpsk (t) and Ĩpsk (t)
The modal decomposition of δis (t):
δis (t) ≈
2r
X
ρjl eλl t +
l=1
2r X
2r
X
ρ0jkl e(λl +λk )t ⇒ Vpsk (t) =
2r
X
αlk eλl t +
l=1
k =1 l=1
2r X
2r
X
α0ijk e(λi +λj )t ,
i=1 j=1
Use Prony to decompose Vpk (t):
Vpk (t) =
N
X
βlk eγl t
l=1
Retain only those modal components within the [0.1,1] Hz. The sum of these
selected modal components are classified as Vpsk (t).
Apply the same procedure to extract θpsk (t), Ipsk (t), and φspk (t)
PMU
PMU
Coherent
Area k
Step 1
Vpk
Coherent
Area k
PMU
18 / 1
Ipk
Step 2
Vpsk
Coherent
Area k
Ipsk
c 2015 by S. Nabavi
Equivalent Area Impedance
KVL in the equivalent circuit:
Eks (t)∠δks (t) = (rks + jxd0sk )Ĩpsk (t) + Ṽpsk (t).
For any time instance:
Φ0 , |(rks + jxd0sk )(Îpsk (t0 )∠φ̂spk (t0 )) + V̂psk (t0 )∠θ̂psk (t0 )|,
..
.
Φm , |(rks + jxd0sk )(Îpsk (tm )∠φ̂spk (tm )) + V̂psk (tm )∠θ̂psk (tm )|.
The estimation of rks and xd0sk can be posed as the following NLS problem:
min var Φ0 , . . . , Φm ,
xd0s , rks
k
rks
Eks  ks
Ipsk
jxdsk
Vpsk  V psk  psk
19 / 1
the kth equivalent
pilot bus
c 2015 by S. Nabavi
Estimating the equivalent generator parameters
Solve the following NLS problem
Z
min
s
s s
Mk ,Dk ,Pm
k
tm
t0
|δ̂ks (t) − δks (t, Mks , Dks , Pms k )|2 dt,
where
δ̇ks (t)
s s
Mk ω̇k (t)
= ωks (t) − ωs ,
s
= Pm
− Pesk − Dis (ωks (t) − 1),
k
δks (t) =δks (t0 ),
ωks (t) = ωks (t0 ), Pesk (t) = Re Êks (t)∠δ̂ks (t) Ĩps∗
(t)
k
rks
Eks  ks
Ipsk
jxdsk
Vpsk  V psk  psk
20 / 1
the kth equivalent
pilot bus
c 2015 by S. Nabavi
Estimating the inter-area impedances
KCL on equivalent pilot buses:
Y s Ṽ s (t0 ) | · · · | Ṽ s (tm ) = Ĩ s (t0 ) | · · · | Ĩ s (tm ) ,
{z
} |
{z
}
|
Ṽ s
Ĩ s
s
Estimate Y by solving:
min
kY s Ṽ s − Ĩ s k2F ,
s
Y
s.t. Y s = (Y s )T
G4s
Ips4
Vps4
G1s
V s
Ips 1 p1
Vps3
Ips3
G3s
Vps2
I s
p2
G2s
21 / 1
c 2015 by S. Nabavi
A Case Study– IEEE 39 Bus Model
.0074+j.0268
9
D4s  1.6541M 4s
Area 4
29
30
22
V ps2
23
20
PMU
4
2
+j
D3s  0
−3
−3
x 10
H 3s  267.2335
G2s
D2s  0.4417 M 2s
G4
−3
1.5
H 2s  82.3485
G7
G5
G3
G2
Area 2
7
5
3
2
0.
05
01
19
35
37
38
.0100+j.0267
.0037+j.0451
13
j 0.0815
24
11
1
59
.0
j0
G1s
6
34
36
-0.0301 -j 1.1996
j0.0058
21
12
G3s
V ps3
7
G6
15
14
33
G1
V ps1
16
97
17
32
.1
j0
39
1+
D1s  0.3227 M 1s
18
PMU
1
V ps4
05
Area 3
PMU
27
H1s  510.6557
j0
.0
81
7
26
.0
-0
31
x 10
1
x 10
1
0.5
(rad/sec)
(rad/sec)
1
0
−0.5
ω̂2s − ω̂1s
ω2s − ω1s
−1
−1.5
0.5
1
1.5
Time (sec)
2
2.5
(rad/sec)
Area 1
PMU
28
0.
00
76
+
25
PMU
17
8
G10
10
H 4s  106.6757
G4s
G9
0.
00
G8
0
−1
−2
0.5
1.5
Time (sec)
22 / 1
2
0
−0.5
ω̂3s − ω̂1s
ω3s − ω1s
1
0.5
2.5
−1
0.5
ω̂4s − ω̂1s
ω4s − ω1s
1
1.5
2
2.5
Time (sec)
c 2015 by S. Nabavi
Future Work
Investigating the utility of the reduced order models for shunt controller
design purposes (such as Static Var Compensator (SVC)).
Vpsk (t )
Gks
Ipsk (t )
Control
Inversion
Coherent
Area k
SVC
SVC
23 / 1
c 2015 by S. Nabavi
Distributed Oscillation Monitoring
Part II– Distributed Optimization Algorithms
for Wide-Area Oscillation Monitoring in
Power Systems
24 / 1
c 2015 by S. Nabavi
Introduction
In Part I, we describe methods to identify
the equivalent models from PMU. In Part
II, we use PMUs to identify the (inter-area)
oscillation modes from PMUs in a
distributed way.
Majority of modal estimation algorithms
are centralized such as: Eigenvalue
Realization Algorithm (ERA)
[Sanchez-Gasca-99], Prony analysis
[Hauer-90], Robust Least Squares
[Zhuo-08], and Hilbert-Huang transform
[Messina-06].
Figure: http://www.eia.gov/
As the number of PMUs scales up into the thousands, the current
state-of-the art centralized architectures will no longer be sustainable.
25 / 1
c 2015 by S. Nabavi
Wide-Area Oscillation Monitoring
Using PMU measurements to estimate the frequency, damping factor and
residue of the different electro-mechanical oscillation modes
G14
PMU
G1
66 40
41
48
47
42
31
38
51
49
G11
52
G16
36
64
37
44
68 43
G13
53
68
6
11
54
19
10
0
5
Time (sec)
23
59
G6
PMU
G2
55
G3
-0.4
-0.5
24
22
58
12
56
G4
-0.3
0.3
0.2
14
13
5
62 G12
0.4
21
7
35
45
39
27
17
15
4
8
61
G9
26
18
9
63
33
34
50
PMU
25
28
16
PMU
G10
46
2
29
G8
56
G15
60
1
30 3
32
62
67
PMU
53
0
5
Time (sec)
26 / 1
G7
20
57
G5
0.24
0.22
0.2
0.18
0.16
0
5
Time (sec)
c 2015 by S. Nabavi
Using •PMU
measurements
to estimate
the the
frequency,
damping
Using
PMU measurements
to estimate
frequency,
damping factor and
residue ofand
theresidue
different
electro-mechanical
oscillation
modes
of the
electro-mechanical oscillation
modes
State-of-the-Art Monitoring
Architecture
The Proposed Distributed
Monitoring Architecture
26 / 1
c 2015 by S. Nabavi
Problem Formulation
Oscillation Monitoring
0
1
1
1
2
2
1
1
2
⋯
⋯
2
0
E1   1
G1
1
1
1
1
2
2
1
2
2
⋯
⋯
~
I L1
V11
E2 2
G2
En  n
Gn
Vn  n
V2  2
~
I L2
~
I L ,n
0
1
Vi i
Gi
Vn 1 n 1
0
1
1
1
1
2
2
2
2
⋯
⋯
1
2
2
2
2
⋯
⋯
G n 1
Ei  i
1
1
1
1
En1 n1
~
I Li
~
I L , n 1
0
1
1
1
1
27 / 1
1
2
2
2
2
⋯
⋯
c 2015 by S. Nabavi
Problem Formulation
Centralized Prony Method
Step 1. Find a1 through a2n
 
∆θi (2n − 1)
···
∆θi (2n)
∆θi (2n)
···
∆θi (2n + 1) 

 
=

.
.
.
.

 
.
.
∆θi (2n + `)
∆θi (2n + ` − 1) · · ·
{z
} |
{z
|

ci
Hi

∆θi (0)
∆θi (1)

. 
. 
.
∆θi (`)
}

−a1
 −a2 


 . 
 . 
.
−a2n
| {z }

a
Finding the global a using all available measurements by solving:
   
c1
H1
.  . 
 .  =  . a
.  . 
Hp
cp
|
{z
}
Solve this using Batch Least Squares - Centralized Prony Method
Step 2. Find the eigenvalues of A (i.e., −σi ± jΩi ) by
– Finding the roots of discrete-time transfer function (z1 through z2n )
– Converting them from discrete-time to continuous-time
28 / 1
c 2015 by S. Nabavi
Problem Formulation
Centralized Prony Method
θi → (Hi , ci ), i = 1, . . . , p
 
 
H1
c1
 
 
⇒  ...  a =  ... 
PMU
G14
41
47
38
67
31
62
51
63
18
G11
35
45
50
39
36
64
37
44
68 43
G13
7
62 G12
6
12
11
10
G2
27
17
21
14
13
5
54

 
H1
c1
1  
 
⇒ a = arg min k  ...  a −  ...  k22
a
2
Hp
cp
G9
15
4
55
G3
19
56
G4
cp

61
26
25
9
8
33
34
52
G16
28
60
16
G10
46
49
32
29
G8
53
2
1
30 3
42
G15
Hp
G1
48
66 40
24
23
22
58
59
G6
G7
20
57
G5
29 / 1
c 2015 by S. Nabavi
Problem Formulation
Distributing the Prony Method
N Computational Areas:
∆θj,i : PMU i in area j
∆θj,i → Hj,i , cj,i
T
T
T
Ĥj , [Hj,1
Hj,2
· · · Hj,N
]T ,
j
G14
48
66 40
41
47
31
62
38
67
51
G11
35
45
39
52
G16
18
36
64
37
44
43
G13
62 G12
21
14
13
5
6
12
11
54
10
G2
Nj : is the total number of PMUs in Area j,
27
17
15
4
7
61
G9
26
25
9
8
33
34
50
68
28
60
16
63
G10
46
49
32
29
G8
53
2
1
30 3
42
G15
T
T
T
· · · cj,N
]T
cj,2
ĉj , [cj,1
j
G1
55
G3
19
56
G4
24
Global Consensus Problem:
23
22
58
59
G6
minimize
G7
20
PN
a1 ,...,aN ,z
57
G5
1
i=1 2 kĤi ai
− ĉi k22
subject to ai − z = 0, for i = 1, . . . , N
Use Alternating Direction Method of Multipliers
(ADMM) to solve it
30 / 1
c 2015 by S. Nabavi
Problem Formulation
Distributing the Prony Method
Three Distributed Cyber-Physical
Architectures (Using ADMM):
Standard ADMM
– Asynchrnous ADMM
G14
G1
48
66 40
41
47
42
31
62
38
67
G15
51
63
18
9
G11
35
45
39
36
64
37
44
43
G13
7
62 G12
6
12
11
10
G2
Distributed ADMM
27
17
21
14
13
5
54
61
G9
15
4
8
33
34
50
G16
28
26
16
52
68
25
1
30 3
32
G10
46
49
60
2
Hierarchical ADMM
29
G8
53
55
G3
19
56
G4
24
23
22
58
59
G6
G7
20
57
G5
30 / 1
c 2015 by S. Nabavi
Distributed Prony Methods
Distributed Prony using Standard ADMM (S-ADMM)
minimize
a1 ,...,aN ,z
PN
1
j=1 2 kĤj aj
Area 1
− ĉj k22
Area 3
PMU
subject to aj − z = 0, for j = 1, . . . , N
θ11(t)
PMU
θ12(t)
θ31(t)
ak
k
1
a
ak
Central PDC
at ISO
k
2
a
ak
θ22(t)
θ21(t)
θ32(t)
k
3
a
a4k
ak
θ42(t)
θ41(t)
PMU
PMU
Area 2
Area 4
Augmented Lagrangian:
Lρ =
N
X
ρ
1
( kĤj aj − ĉj k2 + wjT (aj − z) + kaj − zk2 ),
2
2
j=1
aj , z: the primal variable
wj : the dual variable
ρ: penalty factor
31 / 1
c 2015 by S. Nabavi
Distributed Prony using S-ADMM
Iteration k
Each PDC updates aj locally
ajk +1 = ((Hjk )T Hjk + ρI)−1 ((Hjk )T cjk − wjk + ρz k )
G14
G1
48
66 40
41
47
42
31
62
38
67
G15
49
51
G11
35
45
39
52
G16
32 / 1
28
36
64
37
44
43
G13
62 G12
21
14
13
5
6
12
11
54
10
G2
27
17
15
4
7
61
G9
26
18
9
8
33
34
50
68
25
16
63
G10
46
60
2
1
30 3
32
29
G8
53
55
G3
19
56
G4
24
23
22
58
59
G6
G7
20
57
G5
c 2015 by S. Nabavi
Iteration k
PDC j sends ajk +1 to the central PDC.
G14
G1
48
66 40
41
47
42
31
62
38
67
G15
51
G11
35
45
39
52
G16
32 / 1
28
36
64
37
44
43
G13
62 G12
21
14
13
5
6
12
11
54
10
G2
27
17
15
4
7
61
G9
26
18
9
8
33
34
50
68
25
16
63
G10
46
49
60
2
1
30 3
32
29
G8
53
55
G3
19
56
G4
24
23
22
58
59
G6
G7
20
57
G5
c 2015 by S. Nabavi
Iteration k
The central PDC receives
ajk +1
G14
from all PDCs.
G1
48
66 40
41
31
62
38
67
G15
51
25
28
35
45
52
36
64
37
44
43
G13
62 G12
21
14
13
5
6
12
11
54
10
G2
27
17
15
4
7
61
G9
26
18
9
8
G11
39
G16
60
2
16
63
33
34
50
68
29
G8
53
1
30 3
32
G10
46
49
32 / 1
47
42
55
G3
19
56
G4
24
23
22
58
59
G6
G7
20
57
G5
c 2015 by S. Nabavi
Iteration k
ajk +1
G14
from all PDCs.
G1
31
62
38
67
The central PDC calculates z k +1 =
1
N
PN
k +1
.
j=1 aj
48
66 40
41
G15
49
51
52
G16
25
28
G11
35
45
36
64
37
44
43
G13
62 G12
21
14
13
5
6
12
11
54
10
G2
27
17
15
4
7
61
G9
26
18
9
8
33
34
39
68
60
2
16
63
G10
46
29
G8
53
1
30 3
32
50
32 / 1
47
42
55
G3
19
56
G4
24
23
22
58
59
G6
G7
20
57
G5
c 2015 by S. Nabavi
Iteration k
ajk +1
G14
from all PDCs.
G1
31
62
38
67
1
N
PN
k +1
.
j=1 aj
48
66 40
41
G15
51
32 / 1
52
G16
25
28
G11
35
45
36
64
37
44
43
G13
62 G12
21
14
13
5
6
12
11
54
10
G2
27
17
15
4
7
61
G9
26
18
9
8
33
34
39
68
60
2
16
63
G10
46
49
29
G8
53
1
30 3
32
50
The central PDC sends z k +1 to local PDCs.
47
42
55
G3
19
56
G4
24
23
22
58
59
G6
G7
20
57
G5
c 2015 by S. Nabavi
Iteration k
ajk +1
G14
from all PDCs.
G1
31
62
38
67
1
N
PN
k +1
.
j=1 aj
48
66 40
41
G15
49
51
52
G16
25
28
G11
35
45
36
64
37
44
43
G13
62 G12
21
14
13
5
6
12
11
54
10
G2
27
17
15
4
7
61
G9
26
18
9
8
33
34
39
68
60
2
16
63
G10
46
29
G8
53
1
30 3
32
50
PDC j calculates wjk +1 as
47
42
55
G3
19
56
G4
24
23
22
58
59
G6
G7
20
57
G5
wjk +1 = wjk + ρ(ajk +1 − z k +1 )
32 / 1
c 2015 by S. Nabavi
Iteration k
ajk +1
G14
from all PDCs.
G1
31
62
38
67
1
N
PN
k +1
.
j=1 aj
48
66 40
41
G15
49
51
52
G16
25
28
G11
35
45
36
64
37
44
43
G13
62 G12
21
14
13
5
6
12
11
54
10
G2
27
17
15
4
7
61
G9
26
18
9
8
33
34
39
68
60
2
16
63
G10
46
29
G8
53
1
30 3
32
50
47
42
55
G3
19
56
G4
24
23
22
58
59
G6
G7
20
57
G5
wjk +1 = wjk + ρ(ajk +1 − z k +1 )
The central PDC and local PDCs find the eigenvalues −σi ± jΩi using z k .
32 / 1
c 2015 by S. Nabavi
Distributed Prony using A-ADMM
Iteration k
PDC j sends ajk +1 and wjk to the central PDC.
ajk +1
S k , a subset of PDCs.
The central
P PDC calculates
k +1
z
j∈
/
= N1
S : ajk +1
and
wjk
from
G1
48
66 40
31
62
38
67
G15
51
52
G16
60
2
25
28
9
35
45
36
64
37
44
43
G13
7
21
14
13
5
6
12
11
54
62 G12
10
G2
27
17
15
4
8
G11
61
G9
26
18
16
63
33
34
39
68
29
G8
53
1
30 3
32
50
ajk +1 + ρ1 wjk
ajk , wjk = wjk −1
47
G10
46
49
N
j=1
=
G14
41
42
55
G3
19
56
G4
24
23
22
58
59
G6
G7
20
57
G5
wjk +1 = wjk + ρ(ajk +1 − z k +1 ), j ∈ S k ,
33 / 1
wjk +1 = wjk , j ∈
/ Sk
c 2015 by S. Nabavi
Distributed Prony using H-ADMM
Less communication and
computation overhead for the
central PDC for large number of
PDCs.
Area 3
Area 1
θ11a(t)
The same convergence properties
as the S-ADMM.
θ21a(t)
θ11b(t)
a1k
ak
θ12c(t)
θ22c(t)
θ22a(t)
θ12b(t)
Area 2
34 / 1
θ13a(t)
θ23a(t)
θ13b(t)
ak
ak
θ11c(t)
θ21c(t)
θ12a(t)
θ21b(t)
a2k
θ22b(t)
θ13c(t)
θ23c(t)
a3k
Central PDC
at ISO
a4k
θ14a(t)
θ23b(t)
ak
θ14c(t)
θ24c(t)
θ24a(t) θ14b(t)
θ24b(t)
Area 4
c 2015 by S. Nabavi
Distributed Prony using D-ADMM
Define a communication graph
G(V , E).
Area 1
θ11(t)
A different version of the original
problem defined over G:
minimize
a1 ,...,aN ,z
a1k
a3k w13k
w12k
− ĉj k22
a4k
a1k
a3k
k
14
w
a4k
k
w34
Communication
Graph G
subject to aj − ak = 0, for jk ∈ E(G)
θ21(t)
Modified Augmented Lagrangian:
L0k
ρ =
θ32(t)
θ31(t)
a2k
a
1
j=1 2 kĤj aj
PMU
θ12(t)
k
1
PN
Area 3
PMU
θ41(t)
θ22(t)
θ42(t)
PMU
PMU
Area 2
Area 4
N
X
1X
1
kĤjk aj − ĉjk k2 + ρ(
kavk +1 − aj − wvjk k2 +
2
ρ
j=1
X
v ∈Sj
v ∈Pj
X
1
1 X
kaj − avk − wjvk k2 ) − (
kwvjk k2 +
kwjvk k2 ) ,
ρ
ρ
v ∈Pj
35 / 1
v ∈Sj
c 2015 by S. Nabavi
Resilient Distributed Prony using D-ADMM
One of the challenges of using any distributed computational architecture
is ensuring their resiliency to node attacks in the form of data
manipulation.
It is difficult for the ISO to detect a manipulated set of measurement
broadcasting from a malicious local PDC.
The D-ADMM architecture has the advantage that the primal and dual
updates are done by local PDCs.
Let us define the following residual errors:
Ejk , kĤj ajk − ĉj k,
Ejlk , kĤj alk − ĉj k, ∀l ∈ Nj
let us consider G to be a cycle.
36 / 1
c 2015 by S. Nabavi
1
Each PDC j receives the update of alk +1 for all l ∈ Pj .
37 / 1
c 2015 by S. Nabavi
1
2
PDC j updates aj as ajk +1 = arg min L0ρ .
aj
3
PDC j updates all wlj for l ∈ Pj :
wljk +1
37 / 1
= wljk − ρ(alk +1 − ajk +1 ).
c 2015 by S. Nabavi
1
2
aj
wljk +1
= wljk − ρ(alk +1 − ajk +1 ).
3
4
PDC j sends ajk +1 to all l ∈ Pj ∪ Sj , and receives alk +1 from l ∈ Sj .
37 / 1
c 2015 by S. Nabavi
1
2
aj
wljk +1
= wljk − ρ(alk +1 − ajk +1 ).
3
4
5
PDC j updates all wjl for l ∈ Sj : wjlk +1 = wjlk − ρ(ajk +1 − alk +1 ).
6
PDC j calculates Ejk and Ejlk for l ∈ Pj ∪ Sj .
37 / 1
c 2015 by S. Nabavi
1
2
aj
wljk +1
= wljk − ρ(alk +1 − ajk +1 ).
3
4
5
6
7
If log(Ejlk ) − log(Ejk ) > ET for any l ∈ Pj ∪ Sj , PDC j reports an alert about
node j to the ISO.
37 / 1
c 2015 by S. Nabavi
1
2
aj
wljk +1
= wljk − ρ(alk +1 − ajk +1 ).
3
4
5
6
7
8
If log(Ejlk ) − log(Ejk ) > ET for any l ∈ Pj ∪ Sj , PDC j reports an alert about
node j to the ISO.
If the ISO gets an alert for PDC j from all PDCs belonging to Pj ∪ Sj for K
iterations, it removes PDC j, rearranges a new communication graph G 0
with the remaining PDCs, and continues the iterations.
37 / 1
c 2015 by S. Nabavi
Simulation Results
Simulation Results
A case study for the IEEE 68 bus model,
Area 3
Area 2
G14
PMU
66
41
PMU
48
40
68 bus, 16 generators
Area 1
PMU G8
G1
53
47
60
PMU
26
2
42
PMU
32
3
51
9
15
21
37
44
68
43
G16
5
13
23
22
36 6
PMU
54
64
65
G13
Area 4
24
14
7
35
45
PMU
39
PMU
27
17
4
G11
50
G9
16
8
33
34
52
PMU
18
63
G10
46
49
30
PMU
PMU
G15
1
PMU
31
62
38
67
25
5 computational areas
29
61
28
12
58
11
10
19
56
59
G6 G 7
PMU
G12
PMU
G2
55
G3
G4
Area 5
20
57
PMU
G5
38 / 1
A three-phase fault is
considered occurring at the
line connecting buses 1 and 2.
The fault starts at t = 0.1 sec,
clears at bus 1 at t = 0.15 sec
and at bus 2 at t = 0.20 sec,
Ts =0.2 seconds.
c 2015 by S. Nabavi
Simulation Results
Simulation Results
0.3
0.2
0.6
Ω1
Ω2
Ω3
Ω4
6
5
10
20
30
Iteration (k)
40
2
50
0.4
0.3
0
10
20
30
Iteration (k)
40
0.2
50
0
10
20
30
Iteration (k)
0.4
0.3
40
50
2
0
10
20
30
Iteration (k)
40
50
Ω1
Ω2
Ω3
Ω4
6
5
10 0
S-ADMM
A-ADMM
D-ADMM (G1 )
D-ADMM (G2 )
H-ADMM
10 -6
E
σ
0.5
4
Figure: H-ADMM
7
Ω (rad/sec)
σ1
σ2
σ3
σ4
5
3
Figure: S-ADMM
0.6
Ω1
Ω2
Ω3
Ω4
6
4
3
0
7
σ1
σ2
σ3
σ4
0.5
Ω (rad/sec)
σ
0.4
7
σ
σ1
σ2
σ3
σ4
0.5
Ω (rad/sec)
0.6
4
3
0.2
0.1
0
10
20
30
Iteration (k)
40
50
2
10 -12
0
10
20
30
Iteration (k)
40
50
0
5
10
15
20
25
30
Iteration (k)
35
40
45
50
Figure: A-ADMM
39 / 1
c 2015 by S. Nabavi
Simulation Results
Simulation Results (D-ADMM) with Attack
ET = 8, K = 20.
1010
100
10-10
5
10
15
20
Iteration (k)
10-20
25
-5
10
Error
E12
E15
E1
10-10
30
40
Iteration (k)
50
10
15
20
Iteration (k)
10-20
25
10
E24
E21
E2
10-10
10-15
30
40
Iteration (k)
10-10
5
10-20
25
10-10
30
40
Iteration (k)
5
10
15
20
Iteration (k)
10-5
E45
E42
E4
10-15
50
10
15
20
Iteration (k)
-5
50
25
E51
E54
E5
10-10
10-15
30
40
Iteration (k)
50
0.7
σ1
σ2
σ3
σ4
0.6
Ω1
Ω2
Ω3
Ω4
7
6
0.5
5
Ω
10-15
5
-5
E51
E54
E5
100
10-10
σ
Error
G
10
10-10
Error
10-20
after detection
1010
E45
E43
E4
100
Error
100
1010
E23
E21
E2
PDC 5
Error
E12
E15
E1
Error
G
PDC 4
Error
1010
PDC 2
Error
PDC 1
before detection
4
0.4
3
0.3
2
0.2
10
20
30
Iteration (k)
40
50
40 / 1
10
20
30
Iteration (k)
40
50
c 2015 by S. Nabavi
Simulation Results
Conclusions
Development of distributed algorithms is imperative considering the
increasing number of PMUs in power systems.
We consider the problem of estimating the frequencies and damping
factors of oscillation modes using Prony method in a distributed way.
We proposed three cyber-physical architecture for implementing the
distributed Prony algorithm using several versions of ADMM.
The results of the case studies verify that the distributed solution for the
oscillation modes converges to the centralized solution.
Using a heuristic cross verification method we showed how a malicious
data manipulation can be detected and isolated.
41 / 1
c 2015 by S. Nabavi
Simulation Results
Future Work
Investigating the resiliency of the proposed algorithms under more
complicated attack scenarios.
(Joint work with Jianhua Zhang)
Incorporating the asynchronous wide-area communications considering the
delay traffic models in both uplink and downlink:
Z
t
P(t) =
φ(s)ds =
−∞
1
µ
t −µ
[erf( √ ) + erf( √ )]+
2
2σ
2σ
(p − 1) ( 1 λ2 σ2 +µλ) −λt
λσ 2 + µ
t − λσ 2 − µ
√
e 2
e
[erf( √
) + erf(
)].
2
2σ
2σ
Change the update strategy for downlink (needs convergence proof)
wik = wik −1 + ρ(aik − (z k −1 + γ(z k −1 − z k −2 ))),
42 / 1
i∈
/ S2k
c 2015 by S. Nabavi
Simulation Results
Part III– Graph-Theoretic Identifiability
Analysis of Weighted Consensus Networks
43 / 1
c 2015 by S. Nabavi
Simulation Results
Preliminaries
Consider the following single-input consensus model
defined over a graph G(V , E, W ):
ẋi (t) =
X
wij xj (t) − xi (t) + bi u(t),
i = 1, . . . , n
j∈Ni
Defining x = x1
x2
···
xn
T
ẋ(t) = L(W )x(t) + Bu(t), y(t) = Cx(t), x(0) = 0,
x ∈ Rn , L = −L ∈ Rn×n , B ∈ Rn×1 , W = {wij , ∀ i, j}
[L]i,j

P −wi,j
wi,k
=
 k ∈Ni
0
44 / 1
i∼j
i=j
otherwise
c 2015 by S. Nabavi
Simulation Results
Distinguishability/Identifiability
ẋ(t) = L(W )x(t) + Bu(t), y(t) = Cx(t), x(0) = 0
Consider two distinct parameter sets W and W 0 (W 6= W 0 ). These two
sets are called indistinguishable if the respective models cannot produce
different outputs y(t) for any given input, i.e., y(t, W ) = y(t, W 0 ).
45 / 1
c 2015 by S. Nabavi
Simulation Results
u1 (t )
2.5 
7
2
1
2.5 
7
2
y1 (t )
u2 (t )
2
7
2
2
2
7
2
y2 (t )
Y1 (s)
Y2 (s)
4.5
=
= 4
U1 (s)
U2 (s)
s + 12s3 + 33s2 + 18s
45 / 1
c 2015 by S. Nabavi
Simulation Results
If W and W 0 are not indistinguishable, they are distinguishable.
A parameter set W is said to be globally identifiable if for all W 0 6= W , W
and W 0 are distinguishable.
45 / 1
c 2015 by S. Nabavi
Identifiability Analysis
Identifiability in terms of Markov Parameters
Lemma(Grewel,1976): The parameter sets W and W 0 are
indistinguishable if and only if
CL` (W )B = CL` (W 0 )B, ` ≥ 0.
W is identifiable if and only if the mapping from W to the Markov
parameters is injective (one-to-one).
46 / 1
c 2015 by S. Nabavi
Our Proposition
There are analytical results for identifiability analysis of generic dynamic
models (as early as 70’s and 80’s).
Generally, investigating the parameter identifiability for medium and
large-scale systems is a difficult and intractable task.
We develop a simple sensor placement algorithm to guarantee
identifiability of the edge-weights W for consensus networks defined over
a class of graphs.
We integrate the results from the graph theory with these classical results
of identifiability.
47 / 1
c 2015 by S. Nabavi
Preliminaries
Let us consider a rooted graph G with the root being the input node (node
indexed by 1).
Let us partition V into the following sets:
Si = {v ∈ V : d(v , 1) = i},
i = 0, 1, . . . , p.
S0
S1
S2
1
2



S 2v1


 


v
S22
48 / 1
S 2v3
c 2015 by S. Nabavi
The Studied Class of Graphs
Assumption 1: For a rooted graph G, nodes vl , vq , vs ∈ Si ,
vj ∈ Si+1 , ∀ i ≥ 1 satisfy the following properties:
vl ∼ vj , vq ∼ vj ⇒ vl = vq ,
(vl ∈ Sil ) ∼ (vj ∈ Sij ) ⇒ l = j,
dim({qv ∈ E(G) | q, v ∈ Sij }) ≤ 1, ∀ i, j
49 / 1
c 2015 by S. Nabavi
The Studied Class of Graphs
Assumption 1: For a rooted graph G, nodes vl , vq , vs ∈ Si ,
vj ∈ Si+1 , ∀ i ≥ 1 satisfy the following properties:
vl ∼ vj , vq ∼ vj ⇒ vl = vq ,
(vl ∈ Sil ) ∼ (vj ∈ Sij ) ⇒ l = j,
dim({qv ∈ E(G) | q, v ∈ Sij }) ≤ 1, ∀ i, j
Assumption 2: W is identifiable if C = In .
Parameter b is not identifiable regardless of choice of C.
u (t )
y1 (t )
y2 (t )
49 / 1
y3 (t )
c 2015 by S. Nabavi
Two Supporting Lemmas
Lemma 1– The following holds for L of a G satisfying Assumption 1:
0
0 ≤ k ≤ d(v , 1) − 1
[Lk ]v ,1 =
W(Pv ,1 )
k = d(v , 1)
Pv ,1 is the unique path of length d(v , 1) connecting nodes v and 1.
W(Pv ,1 ) is the weight of path Pv ,1 :
W(P) =
Y
we
e∈P
Proof: By strong induction on k .
50 / 1
c 2015 by S. Nabavi
Two Supporting Lemmas
Lemma 2– Consider a node indexed as v in G and its neighboring nodes
denoted by v1 , . . . , vs . Let L = −L, where L is the weighted Laplacian matrix
of G. If H denotes a subgraph of G induced by the set of all edges incident to
v , and VH and WH denote the vertex set and the weights of all edges
belonging to H respectively, then [Li ]vs ,1 can be uniquely computed from WH
and [Li ]m,1 , (m ∈ VH \{vs }), ∀ i ≥ 1.

vs is called an available node.
51 / 1
c 2015 by S. Nabavi
The Proposed Sensor Placement Algorithm
The Proposed Algorithm
Start with S0 and place a sensor at this node.
52 / 1
c 2015 by S. Nabavi
for k = 1 : p for each set of siblings Skj
choose any |Skj | − 1 nodes belonging to Skj and place sensors at them.
52 / 1
c 2015 by S. Nabavi
for k = 1 : p for each set of siblings Skj
choose any |Skj | − 1 nodes belonging to Skj and place sensors at them.
for each neighboring siblings q, v ∈ Skj −1
q
if Sk and Skv are both non-empty
q
place an additional sensor in either Sk or Skv .
52 / 1
c 2015 by S. Nabavi
The Main Theorem
Consider the following model with G satisfying Assumptions 1 and 2.
If S ⊂ V is a set of sensor nodes determined by the proposed algorithm, y(t)
is the corresponding output measured by S, and H(W ) is the transfer function
from u(t) to y(t), then the mapping from the W to H(W ) is one-to-one.
53 / 1
c 2015 by S. Nabavi
The Main Theorem
Consider the following model with G satisfying Assumptions 1 and 2.
If S ⊂ V is a set of sensor nodes determined by the proposed algorithm, y(t)
is the corresponding output measured by S, and H(W ) is the transfer function
from u(t) to y(t), then the mapping from the W to H(W ) is one-to-one.
Proof:
Wj−1,j , {wu,v ∈ W | u ∈ Sj−1 , v ∈ Sj },
Wj,j , {wu,v ∈ W | u, v ∈ Sj },
j = 1, . . . , p.Qj
, CLj B,
j = 1, 2, . . . , p.
The proof follows from strong induction on j. In each step we proof the
S2n−1
injective mapping of (Wj−1,j , Wj−1,j−1 ) to i=1 Qi .
53 / 1
c 2015 by S. Nabavi
More Results
Proposition 1: If the proposed algorithm is applied to a rooted-tree T ,
then the number of placed sensors is equal to the number of non-input
leaves of T , i.e., the set of leaves that are not the input node.
Proposition 2: If T is a star-graph, then the minimum number of sensors
to identify W is (n − 2).
54 / 1
c 2015 by S. Nabavi
Summary and Conclusions
Summary and Conclusions
We investigate the identifiability problem in Laplacian consensus NDS.
We translate the classical results of identifiability in terms of graph
properties.
We propose a sensor placement algorithm for a class of graphs.
We prove that, our algorithm provides a sufficient condition of
identifiability of the edge weights.
55 / 1
c 2015 by S. Nabavi
Publications
Publications (Published/Accepted)
Book Chapter:
B1 S. Nabavi, J. Zhang, and A. Chakrabortty. Distributed Algorithms for Wide-Area Monitoring in Power
Systems: A Cyber-Physical Perspective. Invited Chapter for CyberPhysical-Social Systems and
Constructs in Electric Power Engineering, IET, 2015.
Journal Articles:
J3 S. Nabavi and A. Chakrabortty, “A Graph-Theoretic Condition for Global Identifiability of Weighted
Consensus Networks," IEEE Transactions on Automatic Control, (conditionally accepted).
J2 S. Nabavi, J. Zhang, and A. Chakrabortty, “Distributed Optimization Algorithms for Wide-Area Oscillation
Monitoring in Power Systems Using Inter-Regional PMU–PDC Architectures," IEEE Transactions on
Smart Grid, vol: p, no: pp, 2015.
J1 T. R. Nudell, S. Nabavi, and A. Chakrabortty, “A Real-Time Attack Localization Algorithm for Large Power
System Networks Using Graph-Theoretic Techniques," IEEE Transactions on Smart Grid, vol: p, no: pp,
2015.
Conference Proceedings:
C5 J. Zhang, S. Nabavi, A. Chakrabortty, and Y. Xin, “Convergence Analysis of ADMM-Based Power System
Mode Estimation Under Asynchronous Wide-Area Communication Delays," IEEE PES General Meeting,
2015 (to be appeared).
C4 S. Nabavi, A. Chakrabortty, and P. P. Khargonekar, “A Global Identifiability Condition for Consensus
Networks with Tree Graphs," American Control Conference (ACC), 2015 (to be appeared).
C3 S. Nabavi and A. Chakrabortty, “Distributed Estimation of Inter-area Oscillation Modes in Large Power
Systems Using Alternating Direction Multiplier Method," IEEE PES General Meeting, National Harbor,
MD, 2014.
C2 S. Nabavi and A. Chakrabortty, “A Real-Time Distributed Prony-Based Algorithm for Modal Estimation of
Power System Oscillations," American Control Conference (ACC), Portland, OR, 2014.
C1 S. Nabavi and A. Chakrabortty, “Topology Identification for Dynamic Equivalent Models of Large Power
System Networks," American Control Conference (ACC), Washington, DC, 2013.
56 / 1
c 2015 by S. Nabavi
Publications
Publications (Under review/ To be submitted)
Journal Articles:
– J. Zhang, S. Nabavi, A. Chakrabortty, and Y. Xin, “ADMM Optimization Strategies for Wide-Area
Oscillation Monitoring in Power Systems under Asynchronous Communication Delays" IEEE
Transactions on Smart Grid - Special Issue on Theory of Complex Systems with Applications to Smart
Grid Operations.
– S. Nabavi and A. Chakrabortty, “Identification of Equivalent DAE Models of Power Systems using
Synchrophasors".
– S. Nabavi and A. Chakrabortty, “Identification of Dynamic Equivalent Models of Power Systems using
Synchrophasors".
Conference Proceedings (submitted):
– S. Nabavi and A. Chakrabortty, “An Intrusion Resilient Distributed Optimization Algorithm for Modal
Estimation in Power Systems", submitted to 2015 IEEE Conference on Control and Decision (CDC).
57 / 1
c 2015 by S. Nabavi

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