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Wide-Area Modeling, Analysis and Control of Large-Scale Power Systems using Synchrophasors

Aranya Chakrabortty

North Carolina State University, Raleigh, NC

Information Trust Institute at UIUC7th October, 2011

Power System Research Consortium (PSRC, 2006-present)

2

Wide Area Measurements (WAMS)• 2003 blackout in the Eastern Interconnection

EIPP (Eastern Interconnection Phasor Project)

NASPI (North American Synchrophasor Initiative)

Industry Members• Rensselaer (Joe Chow, Murat Arcak)• Virginia Tech (Yilu Liu)• Univ. of Wyoming (John Pierre)• Montana Tech (Dan Trudnowski)

• Technical Research (RPI)

1. Model Identification of large-scale power systems 2. Post-disturbance data Analysis3. Controller and observer designs, robustness, optimization

Main trigger: 2003 Northeast Blackout

NYC after blackout

Lesson learnt: 1. Wide-Area Dynamic Monitoring is important2. Clustering and aggregation is imperative

Hauer, Zhou & Trudnowsky, 2004Kosterev & Martins, 2004

3

Ohio New England

INTER-AREASTABLE

INTER-AREA UNSTABLE

Power flow

NYC before blackout

NYC before blackout

NYC after blackout

Lesson learnt: 1. Wide-Area Dynamic Monitoring is important2. Clustering and aggregation is imperative

Ohio New England

INTER-AREASTABLE

INTER-AREA UNSTABLE

Power flow

Hauer, Zhou & Trudnowsky, 2004Kosterev & Martins, 2004

3

Main trigger: 2003 Northeast Blackout

Model Aggregation using distributed PMU data

1. Model Reduction

• How to form an aggregate modelfrom the large system

Problem Formulation:

• Chakrabortty & Chow (2008, 2009, 2010), Chakrabortty & Salazar (2009, 2010)

PMU

PMU

PMU

6-machine, 30 bus, 3 areas

4

1. Model Reduction

• How to form an aggregate modelfrom the large system

Problem Formulation:

Area 1

Area 2 Area 3

Aggregate Transmission

Network

PMU

PMU

PMU

6-machine, 30 bus, 3 areas

• Chakrabortty & Chow (2008, 2009, 2010), Chakrabortty & Salazar (2009, 2010)

Model Aggregation using distributed PMU data

4

111~ VV 222

~ VV

III ~

x1

11 E

H1

xe

22 E

x2

H2

PMU PMU

5

One-Dimensional Models

111~ VV 222

~ VV

III ~

x1

11 E

H1

xe

22 E

x2

H2

sin(

2)21

21

21

2112

21

21

xxxEE

HHPHPH

HHHH

e

mm

Swing Equation

Problem:How to estimate all parameters?

x1, x2, H1, H2

PMU PMU

5

One-Dimensional Models

• Key idea : Amplitude of voltage oscillation at any point is a function of its electrical distance from the two fixed voltage sources.

IME: Method (Reactance Extrapolation)

1E 02EI~

1~V 2

~V

x

1 2

1. Choose a measured variable: Say, voltage magnitude

6


),sin( ] )cos()1([)(~112 aEjaEaExV

21 xxxxae


1E 02EI~

1~V 2

~V

x

1 2

• Voltage magnitude : ,)cos()(2|)(~| 221 aaEEcxVV 2

122

22)1( EaEac

6


),sin( ] )cos()1([)(~112 aEjaEaExV

21 xxxxae

• Voltage magnitude : ,)cos()(2|)(~| 221 aaEEcxVV 2

122

22)1( EaEac


1E 02EI~

1~V 2

~V

x

• Assume the system is initially in an equilibrium (δ0, ω0 = 0, Vss) :

),()( 0aJxV

)sin()(),(

),(:),( 02

0

2100

0

aaaV

EEaVaJ

1 2

6

Reactance Extrapolation

)( )sin(),(),( 20210 aaEEaVtxV

Acan be computed

from measurements at x

(t)

7



A

)( ),( 2aaAtxVn

Note: Spatial and temporal dependence are separated

can be computedfrom measurements at x

(t)

(t)

7



A

)( ),( 2aaAtxVn

Note: Spatial and temporal dependence are separated

can be computedfrom measurements at x

(t)

(t)

*)()( *),( 2 taaAtxVn • Fix time: t=t*

How can we use this relation to solve our problem?

7


1E 02E1 2PMU PMU

*)()( *),( 2 taaAtxVn

8


1E 02E1 2

x2

At Bus 2, *)()( 2222, taaAV Busn

21

22

xxxxa

e

*)()( *),( 2 taaAtxVn

PMU PMU

8


At Bus 1, *)()( 2111, taaAV Busn )1(

)1(1 1

2 2

1,

2,

aaaa

VV

Busn

Busn

1E 02E1 2

x2

xe + x2

21

21

xxxxxa

e

e


21

22

xxxxa

e

• Need one more equation - hence, need one more measurement at a known distance

PMU PMU

*)()( *),( 2 taaAtxVn

8



• Need one more equation - hence, need one more measurement at a known distance )1(

)1(11

33

1,

3,

aaaa

VV

Busn

Busn

1E 02E1 2

x2

xe + x2

21

21

xxxxxa

e

e


21

22

xxxxa

e

3

22

exx

*)()( *),( 2 taaAtxVn

PMU PMU

PMU

)1()1(

1 1

2 2

1,

2,

aaaa

VV

Busn

Busn

8

x1 x2xe/2 xe/2


V1n

V2n

V3n

Vn (a)= A a (1 - a)

Key idea: Exploit the spatial variation of phasor outputs

9

• From linearized model

21

21

HHHHH

where fs is the measured swing frequency and

• For a second equation in H1 and H2, use law of conservation of angular momentum

0 )()(222 221122112211 dtPPPPdtHHHH emem

1

2

2

1

HH

• However, ω1 and ω2 are not available from PMU data,

)(2)cos(

21

21

021

xxxHEE

fe

s

Estimate ω1 and ω2 from the measured frequencies ξ1 and ξ2 at Buses 1 and 2

IME: Method (Inertia Estimation)

• Reminiscent of Zaborsky’s result

1

2

2

1

HH

10

• Express voltage angle θ as a function of δ, and differentiate wrt time to obtain a relation between the machine speeds and bus frequencies:

12111

2121211111 )cos(2

)cos()(cba

cba

22122

2221212122 )cos(2

)cos()(cba

cba

where,

),1( ,)1(22

2

2122

1

ii

iiiii

rEc

rrEEbrEa

• ξ1 and ξ2 are measured, and ai, bi, ciare known from reactance extrapolation.

• Hence, we calculate ω1/ω2 to solve for H1 and H2 .

0 0.2 0.4 0.6 0.8 1-0.4

-0.2

0

0.2

0.4

0.6

Normalized Reactance r

Freq

uenc

y (r/

s)x1

xe x2

IME: Method (Inertia Estimation)

11

Illustration: 2-Machine Example• Illustrate IME on classical 2-machine model (re = 0)• Disturbance is applied to the system and the response simulated in MATLAB

Voltage oscillations at 3 buses

0376.0 0326.0 0301.00136.1 0317.1 0320.10371.0 0316.0 0292.0

321

321

321

nnn

ssssss

mmm

VVVVVVVVV

IME Algorithm

x1 = 0.3382 pux2 = 0.3880 pu

Exact values:x1 = 0.34 pu, x2 = 0.39 pu

1)(

sTssG

Exact values: H1 = 6.5 pu,H2 = 9.5 pu

Bus angle oscillations Bus frequency oscillations

IME H1 = 6.48 pu H2 = 9.49 pu

12

Application to WECC Data

• 2000 gens • 11,000 lines• 22 areas, 6500 loads

Grand Coulee

Colstrip

Vincent

Malin SVC

13

Application to WECC Data

0 50 100 150 200 250 300

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

Time (sec)

Bus

Vol

tage

(pu) Bus 1

Bus 2Midpoint

Needs processing to get usable data

• Sudden change/jump• Oscillations• Slowly varying steady-state (governer

effects)

• 2000 gens • 11,000 lines• 22 areas, 6500 loads

Grand Coulee

Colstrip

Vincent

Malin SVC

14

60 65 70 75 80 85 90-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Time (sec)

Fast

Osc

illatio

ns (p

u)

Bus 1Bus 2Midpoint

0 50 100 150 200 250 3000.9

0.95

1

1.05

1.1

1.15

Time (sec)

Slo

w V

olta

ge (p

u)

0 50 100 150 200 250 300

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

Time (sec)

Bus

Vol

tage

(pu) Bus 1

Bus 2Midpoint

+

Band-passFilter

Oscillations Quasi-steady State

WECC Data

Choose pass-bandcovering typicalswing mode range

14

WECC Data

0 5 10 15 20-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Time (sec)

Inte

rare

a O

scilla

tions

(pu)

Bus 1Bus 2Midpoint

60 65 70 75 80 85 90-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Time (sec)

Fast

Osc

illatio

ns (p

u)

Bus 1Bus 2Midpoint

Oscillations Interarea Oscillations

• Can use modal identification methods such as: ERA, Prony, Steiglitz-McBride

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

x 10-3

Normalized Reactance r

Jaco

bian

Cur

ve

x1x2

xexe

2 2

Bus 1

Bus 2

Bus 3

ERA

15

PMU

PMU

PMUFull Model Aggregated Model

Metrics indicating the dynamic interactionbetween the areas

Aggregated Model

PMU 1

PMU 3PMU 3

Signal Processing

Modes, Amplitude,Residues, Eigenvectors

Application for Stability Assessment

16

Energy Functions for Two-machine System

n

jj

jnn

j

z

j

MdkkSSS

j

ij1

22/)1(

1 *

212

)(

~1 1V

ejx

Gen1 2 2V

~Gen2

Load

P 1 2

'

sin, 2 m

e

E EH P

x

1 2'

sin

e

E EP

x

Kinetic EnergyPotential Energy

)])(sin()cos)[cos(21

opopop

e

(δxEE 2 H

Using IME algorithm: , E1, E2, δ = δ1- δ2, δop, ω = ω1-ω2 & H are computable from

xe, V1, V2, θ = θ1- θ2, θop, ν=ν1-ν2 & ωs

• Note : θop = pre-disturbance angular separation

'ex

17

Energy Functions for WECC Disturbance Event

Sending End and Receiving End Bus Angles

0 50 100 150 200 250 30058

62

66

70

74

Time (sec)

Ang

ular

Diff

eren

ce

(d

eg)

Angle difference between machine internal nodes

IME

Machine speed difference 0 50 100 150 200 250 300

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5x 10

-3

Time (sec)

Mac

hine

Spe

ed D

iffer

ence

(pu)

• Chow & Chakrabortty (2007)

18

Total Energy = Kinetic Energy + Potential Energy

50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3

3.5

Time (sec)

Swin

g C

ompo

nent

of P

oten

tial E

nerg

y ( V

A-s

)

50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3

3.5

Time(sec)

Kin

etic

Ene

rgy

( MW

-s )

50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3

3.5

Time (sec)

Swin

g En

ergy

Fun

ctio

n V

E ( M

W-s

)

Energy Functions for WECC Disturbance Event

• Total energy decays exponentially – damping stability

• Total energy does not oscillate – Out - of - phase osc.– Damped pendulum

Potential Energy

Kinetic Energy

19

Two-Dimensional ModelsMore than Two Areas: Pacific AC Intertie

• Chakrabortty & Salazar (2009, 2010)

• Current in each branch is different• No single spatial variable a• Derivations need to be done piecewise(each edge of the star)

• Two interarea modes/ relative states – δ1 & δ2

Salient Points )()()()( 2211 txJtxJVn

*)()(*)()(*)()(*)()(

24

214

1

23

213

14

3

txJtxJtxJtxJ

VV

n

n

Time-space separation property lost!

20

)()cos()()cos()()sin()()sin()(tan

31211

22111

xfxfxfxfxf

Solution – Use phase angle as a 2nd degree of freedom

)()()()()( 2211 txStxSt

Measurable if a PMU is installed at that point

Pacific AC Intertie

21

)()cos()()cos()()sin()()sin()(tan

31211

22111

xfxfxfxfxf

*)()(*)()(*)()(*)()(

24

214

1

23

213

14

3

txJtxJtxJtxJ

VV

n

n

Solution – Use phase angle as a 2nd degree of freedom

)()()()()( 2211 txStxSt

Measurable if a PMU is installed at that point

*)()(*)()(*)()(*)()(

24

214

1

23

213

14

3

txStxStxStxS

Voltage equation

Pacific AC Intertie

Phase equation

21

Two-Dimensional Models with Direct Connectivity

22

11G12G

13G

21G

22G

23G

31G 32G

33G

Full-Order Model


22

11G12G

13G

21G

22G

23G

31G 32G

33G1G 2G

3G

Must consider ‘Equivalent’ PMU locations

Full-Order Model

Inter-area Model


22

1G 2G

3G

Full-Order Model

Inter-area Model

11

Bi

iBi

i PPII

Current & Power Balance:

ij

Biiji

I

IVV *

*

~

~

1

11G12G

13G

21G

22G

23G

31G 32G

33G

Graph Theoretic Approaches for PMU Placement

11

Bi

iBi

i PPII

Bipartite Graph Problem

• PMU’s should be placed at Minimum vertex cover

23

• Anderson, Chakrabortty and Dobson

Current & Power Balance:

ij

Biiji

I

IVV *

*

~

~

1

• Konig’s theorem for bipartite graphs

• Channel restrictions –We have recently developed new algorithms

D1 D2DT

Area 1 Area 2Transmission Network

Intra-area Network Graph

Intra-area Network Graph

Internal Generators

Internal Generators

NetworkGraph

B1 B2


24

1G 2G

3G

60 65 70 75 80 85 90-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Time (sec)

Fast

Osc

illatio

ns (p

u)

Bus 1Bus 2Midpoint

60 65 70 75 80 85 90-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Time (sec)

Fast

Osc

illatio

ns (p

u)

Bus 1Bus 2Midpoint

Local modes + 2 Interarea modes

Must retain both slow modes in modal decomposition

=ε = xAAAAx )( 3

1421

Not necessarily block‐diagonali.e., slow modes may not be decoupled

2

1

42

31

1

1

)()()()(

uu

sGsGSGsG

yy Off‐diagonal

couplings

In ERA select both slow modes (ignore fast modes), and take impulse response for each PMU bus voltage/phase measurement


24

1G 2G

3G

xAAAAx )( 31

421

What if slow modes are weakly coupled(Almost block‐diagonal)

Lose uniqueness of identifiability

=ε =

2 #

222

22

1 #

112

11

2

ModeInterarea

aa

aa

ModeInterarea

aa

aa

ModesLocal

Ni ii

ii

sss

sss

sss

l

121~z

11

aH

231~z

311~z

12

aH

13

aH

122~z

21

aH

232~z

312~z

22

aH

23

aH

1 #

112

11

ModeInterarea

aa

aa

sss

2 #

222

22

ModeInterarea

aa

aa

sss

• Select either one in ERA, generate separate sets of aggregate model parameters


25

1G 2G

3G

xAAAAx )( 31

421=

ε =

Slow modes are coupled, Identification is unique

Can we still apply IME in a decentralized way?

b

a

IxVE

IxVE~~

~~

2221

1211

When only line a is considered

1

2

a

b

When only line b is considered

Can be matched by adding a fictitious reactance


25

1G 2G

3G

xAAAAx )( 31

421=

ε =

b

a

IxVE

IxVE~~

~~

2221

1211

When only line a is considered

1

2

a

b

When only line b is considered

Can be matched by adding a fictitious reactance (phase-shifting transformer?)

x1

11 E

H1

xe

1~E

1djx

2djx

1Tjxajx

2Tjx

Slow modes are coupled, Identification is unique

Can we still apply IME in a decentralized way?

PMU Placement Problem

• If a tie-line has PMU’s at both ends, what is thebest point of measurement for identification?

Especially if the PMU data are noisy and unreliable?

Model : uI

0

00

L Ej

Want to estimate : rj, xj, H1j, H2j

• For any edge j : )1( )**()()(1

11,

kumAmAakaVn

i

kiji

kijijjj

Spatial variable

26

),()(

),()(

2121 HHkVK

xxkVH

TT

TT

KKKHHKHH

J

Fisher Information Matrixdepends on aj

• Stack up measurements & define :

• Problem statement: Find optimal aj s.t. Cramer-Rao Bound is minimised

The PMU Allocation Problem

• Chakrabortty & Martin (2010, 2011)

1 2 3 4 5 6 7 8 9 100.955

0.956

0.957

0.958

0.959

0.96

0.961

0.962

0.963

0.964

0.965

Time (sec)

Voltage

Mag

nitude

(pu)

Noisy dataExtracted Modal Response

0 1 2 3 4 5 6 7 8 9 101.7

1.75

1.8

1.85

1.9

1.95

2

Time (sec)

Voltage

Mag

nitude

(pu)

Noisy DataExtracted Modal Response

Bus 2 voltage

Bus 1 voltage

0.1 0.2 0.3 0.4 0.5

0

2

4

6

8

10

12

14x 10-3

Reactance (pu)

Nor

mal

ized

Det

erm

inan

t of F

IM

Bus 3Bus 13

xe

Spatial variation of the determinant of the FIM

26

Parameters

ActualValues

Iteration 1 Iteration 1 Iteration 1 Iteration 1 Iteration 1

r 0.1 0.05 0.06 0.08 0.08 0.093

x 1 0.58 0.64 0.78 0.83 0.981

H1 19 15.65 17.91 17.65 18.55 18.98

H2 13 10.86 10.91 11.68 12.35 12.75

a - 0.428 0.412 0.409 0.408 0.408

27

PMU based Disturbance Modeling

Frequency Distribution captured by FNET-FDRs

Swing Equation becomes a PDE

Hauer (1983), Thorp & Parashar (2003)

xc

t 2

22

2

2

(Source behind the

Frequency wavesin FNET)

27

PMU based Disturbance Modeling

Frequency Distribution captured by FNET-FDRs

Swing Equation becomes a PDE

Hauer (1983), Thorp & Parashar (2003)

xc

t 2

22

2

2

(Source behind the

Frequency wavesin FNET)

How to model for a network with given topology using

PMUs ?

The Wide-area Control Problem• Computational complexity sharply increases with number of areas

PMU PMU

PMU PMU

Area Identification Border Buses

Network Reduction *Control Allocation-

Inverse Problem

28

The Cyber-Physical Challenge• Distributed Identification/Simulation:

A number of computers solve assigned chunks of the system dynamics and Exchange information to update the state - coupling

xAAAA

xXX

xxxx

xAxx

xxxx

x

43

21

2

1

4

3

2

1

4

3

2

1

, ,

1A 4A

2A 3A

29

The Cyber-Physical Challenge• Distributed Identification/Simulation:

A number of computers solve assigned chunks of the system dynamics and Exchange information to update the state - coupling

xAAAA

xXX

xxxx

xAxx

xxxx

x

43

21

2

1

4

3

2

1

4

3

2

1

, ,

1A 4A

2A 3A

Exchange will depend on the connection graph

Actuation: Frequency feedbackFACTS: STATCOM, SVC

29

Distributed Architecture for PMU Applications

State-of-art Centralized Processing

Distributed PMU Networks

30

Optimization variables: Data exporting rates of PMUs or virtual PMUsInformation update rate between PDCs – local and inter-areaNetwork bandwidth

Constraints: Network delays, Processing delays, Congestion, Dynamic routing

• Define an execution metric M for each application of the PMU-net

• Minimize the error between distributed Md and corresponding centralized Mc

Dynamic Rate Control Problem(DRCP)

Dynamic Link Assignment Problem (DLAP)

31

How about setting up anintra-campus local PMUcommunication networkwith RENCI/UNC and Duke Univ.?

Phasor Lab

1. Real PMU Data from WECC (NASPI data)2. RTDS-PMU Data (Schweitzer PhasorLab)3. FACTS-TNA with NI CRIO PMUs

We can provide all three data via our new PMU and RTDS facilities at the FREEDMcenter

32

TR - SH200 MVA

INVERTER 1100 MVA

DC Bus 1 DC Bus 2

SWDC

TBS1

LVCB

HSB

HSB

INVERTER 2100 MVA

Marcy 345 kVNorth Bus

Marcy 345 kVSouth Bus

Edic

Volney AT1 AT2

TBS2

LVCB

TR - SE1100 MVA

TR - SE2100 MVA

NewScotland

(UNS)

CoopersCorners(UCC)

HVCB

Transient Network Analyzer

• Joint work with Subhashish Bhattacharya (NYPA, Siemens)• Real-time emulation of the NY grid• Create fault injections at vulnerable points, measure via 3 PMUs from National Instruments• Ideal test-bed for small-scale dynamic visualization within NY state• More ambitious – controller tuning for damping control

www.ece.ncsu.edu/power33

Conclusions1. WAMS is a tremendously promising technology for smart grid researchers

2. Communications and Computing must merge with power engineering

3. Plenty of new research problems – EE, Applied Math, Computer Science

4. Cyber-security is essential

5. Right time to think mathematically – Network theory is imperative

6. Right time to pay attention to the bigger picture of the electric grid

7. Needs participation of young researchers!

8. Promises to create jobs and provide impetus to the ARRA

1. Funding support provided by NSF, SCE2. PMU data and software provided by SCE, BPA, and EPG

Acknowledgements:

34

Thank YouEmail: [email protected]

Homepage: http://people.engr.ncsu.edu/achakra2

PMU Lab: www.ece.ncsu.edu/power

35

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