distributed estimation of inter-area oscillation modes in ... · 3/12/2014 · modes in large...

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

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Time (sec)

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Time (sec)

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0 5

0.160.180.2

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Time (sec)

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S. Nabavi and A. Chakrabortty 4/19


State of the Art Monitoring Architecture

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PMU



State of the Art Monitoring Architecture Proposed Distributed Monitoring Architecture

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PMU

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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


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)


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.



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



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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


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


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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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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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


Simulation Results

Simulation results for the IEEE-39 bus model,

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2

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2724

12

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10

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5

7

8

932 34 33

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19 23

4

3

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22

35

PMU

PMU

PMU

PMU

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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).


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


• 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.


distributed estimation of inter-area oscillation modes in ... · 3/12/2014 · modes in large...

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