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Distributed Estimation of Inter-Area Oscillation Modes in Large Power Systems
using ExoGENI-WAMS Communication Network
Aranya Chakrabortty FREEDM Systems Center
North Carolina State University, Raleigh
March 12, 2014 NASPI Working Group Meeting, Knoxville, TN
Introduction
Problem Formulation - Wide Area Oscillation Monitoring
Centralized Prony Method
Distributed Prony Method
Simulation Results
Conclusions and Future Work
Wide-Area Oscillation Monitoring
Using PMU measurements to estimate the frequency, damping factor and residue of thedifferent electro-mechanical oscillation modes
IEEE 68-Bus Model The WECC Model
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PMU
PMUPMU
PMU
PMU
0 50.2
0.3
0.4
Time (sec)
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0 5-0.5
-0.4
-0.3
Time (sec)
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0 5
0.160.180.2
0.220.24
Time (sec)
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S. Nabavi and A. Chakrabortty 4/19
Wide-Area Oscillation Monitoring
State of the Art Monitoring Architecture
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PMU
S. Nabavi and A. Chakrabortty 5/19
Wide-Area Oscillation Monitoring
State of the Art Monitoring Architecture Proposed Distributed Monitoring Architecture
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PMU
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S. Nabavi and A. Chakrabortty 5/19
Wide-Area Oscillation Monitoring
State of the Art Monitoring Architecture Proposed Distributed Monitoring Architecture
Pros:• Less Communication• Guaranteed data privacy
Cons:• High risk for security and resiliency• High computational load for central
computer• Higher computational time for very
large data volumes
Pros:• Reduced computational time• Privacy still preserved• More secure and resilient• More efficient and tractable data
handling
Cons:• Significant increase in communication
infrastructure• Asynchrony between PDCs• Communication delays
S. Nabavi and A. Chakrabortty 5/19
Swing Equation
• Swing equation of the ith machine:
δi = ωs(ωi − 1)
Miωi = Pmi −∑k
(EiEk
xiksin(δik)
)−Di(ωi − 1)
• Linearized dynamic model (after Kron reduction)[∆δ(t)
∆ω(t)
]=
[0 ωsIn
M−1L −M−1D
]︸ ︷︷ ︸
A
[∆δ(t)
∆ω(t)
]+
[0
M−1ej
]︸ ︷︷ ︸
B
u(t),
y(t) = col(∆θ), for i ∈ S
Lii = −∑k∈Ni
EiEk
xikcos(δi0 − δk0), Lij =
EiEj
xijcos(δi0 − δj0)
1
~LI1G
Bus 1 11 V
2G
iG1nG
nGBus 2
22 V
Bus iiiV
Bus n-111 nnV
nnV
Bus n
2
~LI
LiI~
1,
~nLI
nLI ,
~
11 E
22 E
iiE 11 nnE
nnE
Figure: A power system with bothPV buses (differential bus) and PQbuses (algebraic bus)
S. Nabavi and A. Chakrabortty 6/19
Oscillation Monitoring
yj(t) = ∆θj(t) =n∑i=1
(rj,ie
(−σi+jΩi)t + r∗j,ie(−σi−jΩi)t
)
y(t) =
∆θ1(t)...
∆θp(t)
=n∑i=1
(r1,i
...rp,i
e(−σi+jΩi)t +
r∗1,i
...r∗p,i
e(−σi−jΩi)t)
• Our objective is to use PMU measurements y(t) to estimate Ωi, σi andcol(r1,i, . . . , rp,i) for i = 1, · · · , n
• We use Prony algorithm for this.• Let us consider the discrete-time transfer function of ∆θi from a single input
disturbance:
∆θi(z) =b0 + b1z−1 + b2z−2 + · · ·+ b2nz−2n
1 + a1z−1 + a2z−2 + · · ·+ a2nz−2n.
S. Nabavi and A. Chakrabortty 7/19
Centralized Prony Method
Step 1. Find a1 through a2n∆θi(2n)
∆θi(2n+ 1)...
∆θi(2n+ `)
︸ ︷︷ ︸
ci
=
∆θi(2n− 1) · · · ∆θi(0)
∆θi(2n) · · · ∆θi(1)...
...∆θi(2n+ `− 1) · · · ∆θi(`)
︸ ︷︷ ︸
Hi
−a1
−a2
...−a2n
︸ ︷︷ ︸
a
Finding the global a using all available measurements by solving:c1
...cp
=
H1
...Hp
a
︸ ︷︷ ︸Solve this using Batch Least Squares - Centralized Prony Method
S. Nabavi and A. Chakrabortty 8/19
Centralized Prony Method
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PMU
θi → (Hi, ci), i = 1, . . . , p
⇒
H1
...Hp
a =
c1
...cp
⇒ a = arg mina
1
2‖
H1
...Hp
a−
c1
...cp
‖22
S. Nabavi and A. Chakrabortty 10/19
Distributing the Prony Method
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Multiple Computational Areas
θ1 , θ30, θ66 → (H1 ,
[H30
H66
], c1 ,
[c30
c66
])
θ2 , θ16, θ53 → (H2 ,
[H16
H53
], c2 ,
[c16
c53
])
θ3 , θ68 → (H3 , H68, c3 , c68)
θ4 , θ56 → (H4 , H56, c4 , c56)
Global Consensus Problem:
minimizea1,...,aN ,z
∑Ni=1
12‖Hiai − ci‖22
subject to ai − z = 0, for i = 1, . . . , N
S. Nabavi and A. Chakrabortty 11/19
Distributed Optimization Algorithms
• Gradient-Based Methods• Distributed Subgradient Method
(DSM)• Nesterov Method
• Dual Decomposition Based Methods• Alternating Direction Method of
Multipliers (ADMM)
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S. Nabavi and A. Chakrabortty 12/19
Alternating Direction Method of Multipliers (ADMM)
Iteration k
Step 1 Update wi and ai locally at PDC i
a(k+1)i = ((H
(k)i )TH
(k)i + ρI)−1((H
(k)i )T c
(k)i −w
(k)i + ρa(k))
w(k+1)i = w
(k)i + ρ(a
(k+1)i − a(k+1))
Step 2 Gather the values of a(k+1)i at the central PDC
Step 3 Take the average of a(k+1)i
Step 4 Broadcast the average value (a(k+1)) to local PDCs
Step 5 Finding the frequency Ωi and damping factors σi at eachlocal PDC using a(k) from the characteristic equation
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S. Nabavi and A. Chakrabortty 13/19
Targeted Estimation of Inter-Area Modes
Given PMU data y(t) = col(∆Vi,∆θi), we can estimatemodes of the aggregated model.
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
0 .9 4
0 .9 6
0 .9 8
1
1 .0 2
1 .0 4
1 .0 6
1 .0 8
1 .1
T im e (s e c )
Bu
s V
olta
ge
(p
u)
B u s 1B u s 2M id p o in t
Time (sec)
Bus 1Bus 2Bus 3
Time (sec)
• Massive volumes of PMU data from various buses
• Assuming inter-area modes to lie between 0.1 Hz and 1 Hz, applyband-pass filtering
• Use filtered data to estimate oscillations between aggregateclusters
CanadaWashington
Oregon
Malin
Pacific AC Intertie
Table Mountain
Pacific AC Intertie
Vincent
Montana Wyoming
Utah
ArizonaNew Mexico
Pacific HVDCIntertie
LugoLos Angeles
Baja CA (Mexico)
IntermountainHVDCLine
PMU
PMU PMU
PMU
PMU
S. Nabavi and A. Chakrabortty 14/19
Simulation Results
Simulation results for the IEEE-39 bus model,
30 37
25
26 2829
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2
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2724
12
6
1113
10
31
5
7
8
932 34 33
3620
19 23
4
3
39
16
1514 21
22
35
PMU
PMU
PMU
PMU
G1
G10 G8
G3G2 G4G5
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• Simplified model of theNew-England power system
• 39 Bus, 10 Generators• 4 Coherent Areas (shown in
different colors)• Simulations are performed in
Power System Toolbox (PST)• A three-phase fault occurred at line
connecting buses 4 and 5, started att = 1.0 (sec), cleared at near end att = 1.01 (sec), and cleared at farend at t = 1.03 (sec).
S. Nabavi and A. Chakrabortty 15/19
Simulation Results
0 50 100−5.5
−5
−4.5
−4
−3.5
Iteration (k)
log 10
(ei)
i=1i=2i=3i=4
0 50 1000
0.5
1
1.5
Iteration (k)
Dam
ping
Fac
tor
σ1
σ2
σ3
0 50 1002
4
6
8
10
Iteration (k)
Inte
r−A
rea
Fre
quen
cy (
rad/
sec)
Ω1
Ω2
Ω3
Actual Eigenvalues Centralized Prony Decentralized Prony Distributed Prony−0.2333± 3.7128j −0.2341± 3.7127j −0.2339± 3.7142j −0.2339± 3.7124j−0.2375± 5.7914j −0.2323± 5.7638j −0.3818± 5.5658j −0.2199± 5.7673j−0.2718± 6.4277j −0.3014± 6.4228j −0.2928± 6.3887j −0.3003± 6.4224j
S. Nabavi and A. Chakrabortty 16/19
• RTDS: Simulate high fidelity detailed models of large power systems • MS: Multi-ventor PMU-based hardware-in-loop simulation testbed • ExoGENI: Widely distributed networked IaaS platform for experimentation and computational
tasks. • PDCs connected to ExoGENI network through 10 Gbps Breakable Experimental Network (BEN).
Implementation via Distributed Exo-GENI Communication Network
RTDS-PMU Lab at NCSU Fiber optic network connecting campuses of NC State, Duke University & UNC Chapel Hill
Experimental Network Topologies
Centralized Decentralized (independent) Followed by
averaging of estimates
Decentralized but recursive
Calculating End-to-End Network Delays
Calculating End-to-End Network Delays
Choice of VM location decided by network traffic
Conclusions
• Development of distributed algorithms is imperative considering the increasingnumber of PMUs.
• We consider the problem of estimating the frequencies and damping factors ofoscillation modes using Prony method in a distributed way.
• The results of ADMM verify that the global values of the inter-area modes can beachieved after a number of iterations.
S. Nabavi and A. Chakrabortty 18/19
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