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SLIDE 1

A Measurement-based Framework for Modeling, Analysis and Control of Large-Scale Power Systems using Synchrophasors

Aranya Chakrabortty

Electrical & Computer Engineering Department North Carolina State University LCCC Workshop, Lund University 18th May, 2011

SLIDE 2

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 Analysis
• 3. Controller and observer designs, robustness, optimization

SLIDE 3

NYC before blackout NYC after blackout

Lesson learnt:

• 1. Wide-Area Dynamic Monitoring is important
• 2. Clustering and aggregation is imperative

Ohio New England INTER-AREA STABLE INTER-AREA UNSTABLE

Power flow Hauer, Zhou & Trudnowsky, 2004 Kosterev & Martins, 2004 3

Main trigger: 2003 Northeast Blackout

SLIDE 4

Model Aggregation using distributed PMU data

• 1. Model Reduction
• How to form an aggregate model

from the large system Problem Formulation:

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

PMU PMU PMU

6-machine, 30 bus, 3 areas

4

SLIDE 5

• 1. Model Reduction
• How to form an aggregate model

from 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

SLIDE 6

1 1 1

~    V V

2 2 2

~    V V

I

I I    ~

x1

1 1

  E

H1 xe

2 2

  E

x2 H2

PMU PMU 5

Two-area Model Estimation

SLIDE 7

Two-area Model Estimation

1 1 1

~    V V

2 2 2

~    V V

I

I I    ~

x1

1 1

  E

H1 xe

2 2

  E

x2 H2

    sin ( 2

) 2 1 2 1 2 1 2 1 1 2 2 1 2 1

x x x E E H H P H P H H H H H

e m m

         

Swing Equation Problem: How to estimate all parameters? x1, x2, H1, H2

PMU PMU 5

SLIDE 8

• Key idea : Amplitude of voltage oscillation at any point is a function of its electrical

distance from the two fixed voltage sources.

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

1 1 2

  a E j a E a E x V    

2 1

x x x x a

e 

 

IME: Method (Reactance Extrapolation)

 

1

E

2

E

I ~

1

~ V

2

~ V

x

1 2

• Voltage magnitude :

, ) cos( ) ( 2 | ) ( ~ |

2 2 1

 a a E E c x V V    

2 1 2 2 2 2

) 1 ( E a E a c   

6

SLIDE 9

• Key idea : Amplitude of voltage oscillation at any point is a function of its electrical

distance from the two fixed voltage sources.

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

1 1 2

  a E j a E a E x V    

2 1

x x x x a

e 

 

• Voltage magnitude :

, ) cos( ) ( 2 | ) ( ~ |

2 2 1

 a a E E c x V V    

2 1 2 2 2 2

) 1 ( E a E a c   

IME: Method (Reactance Extrapolation)

 

1

E

2

E

I ~

1

~ V

2

~ V

x

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

     ) , ( ) ( a J x V

) sin( ) ( ) , ( ) , ( : ) , (

2 2 1

    

 

a a a V E E a V a J      



1 2

6

SLIDE 10

Reactance Extrapolation

        ) ( ) sin( ) , ( ) , (

2 2 1

a a E E a V t x V

A

    ) ( ) , (

2

a a A t x V n

Note: Spatial and temporal dependence are separated can be computed from measurements at x

(t)

(t)

*) ( ) ( *) , (

2

t a a A t x V n    

• Fix time: t=t*

How can we use this relation to solve our problem?

7

SLIDE 11

Reactance Extrapolation

At Bus 1,

*) ( ) (

2 1 1 1 ,

t a a A V

Bus n

   

) 1 ( ) 1 (

1 1 2 2 1 , 2 ,

a a a a V V

Bus n Bus n

  

 

1

E

2

E

1 2

x2

xe + x2

2 1 2 1

x x x x x a

e e

   

At Bus 2,

*) ( ) (

2 2 2 2 ,

t a a A V

Bus n

   

2 1 2 2

x x x x a

e 

 

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

PMU PMU

*) ( ) ( *) , (

2

t a a A t x V n    

8

SLIDE 12

Reactance Extrapolation

At Bus 1,

*) ( ) (

2 1 1 1 ,

t a a A V

Bus n

   

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

) 1 ( ) 1 (

1 1 3 3 1 , 3 ,

a a a a V V

Bus n Bus n

  

 

1

E

2

E

1 2

x2

xe + x2

2 1 2 1

x x x x x a

e e

   

At Bus 2,

*) ( ) (

2 2 2 2 ,

t a a A V

Bus n

   

2 1 2 2

x x x x a

e 

 

3

2

2 e

x x 

*) ( ) ( *) , (

2

t a a A t x V n    

PMU PMU PMU

) 1 ( ) 1 (

1 1 2 2 1 , 2 ,

a a a a V V

Bus n Bus n

  

8

SLIDE 13

• From linearized model

2 1 2 1

H H H H H  

where fs is the measured swing frequency and

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

) ( ) ( 2 2 2

2 2 1 1 2 2 1 1 2 2 1 1

       

 

dt P P P P dt H H H H

e m e m

     

1 2 2 1

    H H

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

) ( 2 ) cos( 2 1

2 1 2 1

x x x H E E f

e 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

    H H 9

SLIDE 14

• Express voltage angle θ as a function of δ, and differentiate wrt time to obtain a

relation between the machine speeds and bus frequencies:

1 2 1 1 1 2 1 2 1 2 1 1 1 1 1

) cos( 2 ) cos( ) ( c b a c b a                 

2 2 1 2 2 2 2 2 1 2 1 2 1 2 2

) cos( 2 ) cos( ) ( c b a c b a                 

where,

), 1 ( , ) 1 (

2 2 2 2 1 2 2 1 i i i i i i i

r E c r r E E b r E a     

• ξ1 and ξ2 are measured, and ai, bi, ci

are known from reactance extrapolation.

• Hence, we calculate ω1/ω2 to solve

for H1 and H2 .

0.2 0.4 0.6 0.8 1

• 0.4
• 0.2

0.2 0.4 0.6 Normalized Reactance r Frequency (r/s) x1 xe x2

IME: Method (Inertia Estimation)

10

SLIDE 15

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 . 0326 . 0301 . 0136 . 1 0317 . 1 0320 . 1 0371 . 0316 . 0292 .

3 2 1 3 2 1 3 2 1

        

n n n ss ss ss m m m

V V V V V V V V V IME Algorithm x1 = 0.3382 pu x2 = 0.3880 pu Exact values: x1 = 0.34 pu, x2 = 0.39 pu

1 ) (   sT s s G

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

11

SLIDE 16

Application to WECC Data

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

Grand Coulee Colstrip Vincent Malin SVC

12

SLIDE 17

Application to WECC Data

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 Voltage (pu) Bus 1 Bus 2 Midpoint

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

12

SLIDE 18

60 65 70 75 80 85 90

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02 Time (sec) Fast Oscillations (pu) Bus 1 Bus 2 Midpoint 50 100 150 200 250 300 0.9 0.95 1 1.05 1.1 1.15 Time (sec) Slow Voltage (pu) 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 Voltage (pu) Bus 1 Bus 2 Midpoint

+

Band-pass Filter

Oscillations Quasi-steady State

WECC Data

Choose pass-band covering typical swing mode range 13

SLIDE 19

WECC Data

5 10 15 20

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02 Time (sec) Interarea Oscillations (pu) Bus 1 Bus 2 Midpoint 60 65 70 75 80 85 90

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02 Time (sec) Fast Oscillations (pu) Bus 1 Bus 2 Midpoint

Oscillations Interarea Oscillations

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

0.2 0.4 0.6 0.8 1 5 10 15 20 x 10

• 3

Normalized Reactance r Jacobian Curve

x1 x2 xe xe 2 2 Bus 1 Bus 2 Bus 3

ERA

14

SLIDE 20

PMU PMU PMU

Full Model Aggregated Model

Metrics indicating the dynamic interaction between the areas Aggregated Model

PMU 1 PMU 3 PMU 3

Signal Processing

Modes, Amplitude, Residues, Eigenvectors

Application for Stability Assessment

15

SLIDE 21

Energy Functions for Two-machine System

  

  

   

n j j j n n j z j

M dk k S S S

j ij

1 2 2 / ) 1 ( 1 * 2 1

2 ) (  



~

1 1

V  

e

jx Gen1

2 2

V  

~

Gen2 Load P

 

1 2 '

sin , 2

m e

E E H P x        

 

1 2 '

sin

e

E E P x  

Kinetic Energy Potential Energy

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

2 1

        

• p
• p
• p

e

(δ x E E

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

'

e

x

16

SLIDE 22

Energy Functions for WECC Disturbance Event

Sending End and Receiving End Bus Angles

50 100 150 200 250 300 58 62 66 70 74 Time (sec) Angular Difference  (deg)

Angle difference between machine internal nodes

IME

Machine speed difference

50 100 150 200 250 300

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 2.5 x 10

• 3

Time (sec) Machine Speed Difference (pu)

• Chow & Chakrabortty (2007)

17

SLIDE 23

Total Energy = Kinetic Energy + Potential Energy

50 100 150 200 250 300 0.5 1 1.5 2 2.5 3 3.5 Time (sec) Swing Component of Potential Energy ( VA-s ) 50 100 150 200 250 300 0.5 1 1.5 2 2.5 3 3.5 Time(sec) Kinetic Energy ( MW-s )

50 100 150 200 250 300 0.5 1 1.5 2 2.5 3 3.5 Time (sec) Swing Energy Function VE ( MW-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

18

SLIDE 24

More 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

) ( ) ( ) ( ) (

2 2 1 1

t x J t x J Vn      

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

2 4 2 1 4 1 2 3 2 1 3 1 4 3

t x J t x J t x J t x J V V

n n

          

Time-space separation property lost!

19

SLIDE 25

           



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

3 1 2 1 1 2 2 1 1 1

x f x f x f x f x f      *) ( ) ( *) ( ) ( *) ( ) ( *) ( ) (

2 4 2 1 4 1 2 3 2 1 3 1 4 3

t x J t x J t x J t x J V V

n n

          

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

) ( ) ( ) ( ) ( ) (

2 2 1 1

t x S t x S t        

Measurable if a PMU is installed at that point

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

2 4 2 1 4 1 2 3 2 1 3 1 4 3

t x S t x S t x S t x S               

Voltage equation

Pacific AC Intertie

Phase equation

20

SLIDE 26

PMU Placement Problem

• If a tie-line has PMU’s at both ends, what is the

best point of measurement for identification?

Especially if the PMU data are noisy and unreliable? Model : u I                                    

L Ej

Want to estimate : rj, xj, H1j, H2j

21

SLIDE 27

PMU Placement Problem

• If a tie-line has PMU’s at both ends, what is the

best point of measurement for identification?

Especially if the PMU data are noisy and unreliable? Model : u I                                    

L Ej

Want to estimate : rj, xj, H1j, H2j

• For any edge j :

) 1 ( ) * * ( ) ( ) (

1 1 1 ,

  



  

k u m A m A a k a V

n i k i ji k i ji j j j



Spatial variable

21

) , ( ) ( ) , ( ) (

2 1 2 1

H H k V K x x k V H             

T T T T

KK KH HK HH J

Fisher Information Matrix depends on aj

• Stack up measurements & define :
• Problem statement: Find optimal aj s.t. Cramer-Rao Bound is minimised

SLIDE 28

The PMU Allocation Problem

• Chakrabortty & Sczodrak (2010)

1 2 3 4 5 6 7 8 9 10 0.955 0.956 0.957 0.958 0.959 0.96 0.961 0.962 0.963 0.964 0.965 Time (sec) Voltage Magnitude (pu) Noisy data Extracted Modal Response 1 2 3 4 5 6 7 8 9 10 1.7 1.75 1.8 1.85 1.9 1.95 2 Time (sec) Voltage Magnitude (pu) Noisy Data Extracted Modal Response

Bus 2 voltage Bus 1 voltage

0.1 0.2 0.3 0.4 0.5 2 4 6 8 10 12 14x 10

• 3

Reactance (pu) Normalized Determinant of FIM

Bus 3 Bus 13 xe

Spatial variation of the determinant of the FIM

22

Parameter s Actual Values 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

SLIDE 29

The Wide-area Control Problem

• Computational complexity sharply increases with number of areas
• Chakrabortty, Nudell & Martin (ongoing)

Area Identification Border Buses Network Reduction * 23

SLIDE 30

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 23

SLIDE 31

12 12 12

~ x j r z  

1

H

2

H

3

H

4

H

5

H

6

H

1 4

6

2 3 5

8 7

45 45 45

~ x j r z  

1

H

2

H

3

H

4

H

5

H

6

H 1 4

6 2

3

5 8 7

Area 1 Area 2 Area 3 Area 1 Area 2 Area 3

Inter-area links Boundary node for Area 2 & 3 Boundary nodes for Area 1

eq

z

, 12

~

eq

z

, 23

~

eq

z

, 31

~

1

H

2

H

3

H

4

H

5

H

6

H

1 4

6 2

3

5 8 7

Area 1 Area 2 Area 3

                    

2 # 2 2 2 2 2 1 # 1 1 2 1 1 2 Mode Interarea a a a a Mode Interarea a a a a Modes Local N i i i i i

s s s s s s s s s

l

                      



 12 1

~ z

1 1 a

H

23 1

~ z

31 1

~ z

1 2 a

H

1 3 a

H

12 2

~ z

2 1 a

H

23 2

~ z

31 2

~ z

2 2 a

H

2 3 a

H

      

1 # 1 1 2 1 1 Mode Interarea a a a a

s s s              

2 # 2 2 2 2 2 Mode Interarea a a a a

s s s       

) ( ) ( ) (

1 1 1 1 2 1 1 l u l u l u

s s s          

Band Pass Filter # 1

) ( ) ( ) (

2 2 2 2 2 2 2 l u l u l u

s s s          

Band Pass Filter # 2

Control depends on the choice of Interarea Mode:

1 1 1

~    V V

2 2 2

~    V V

I

I I    ~ x1

1 1

  E

H1 xe

2 2

  E

x2 H2

PMU PMU

Two-area system with 1 dominant Interarea mode

24

SLIDE 32

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

• 0.2

0.2 0.4 0.6 0.8 1 1.2 Time (sec) V1-V2 (pu) 2-area Interarea 2-machine 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.2 0.4 0.6 0.8 1 1.2 1.4 Time (sec) V1-V2 2-area Interarea 2-machine

PMU PMU

2-area Kundur system

Band-pass filter with relative degree zero

25

SLIDE 33

The Cyber-Physical Challenge

26

• Distributed Identification/Simulation:

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

x A A A A x X X x x x x x Ax x x x x x x                                         

4 3 2 1 2 1 4 3 2 1 4 3 2 1

, ,  

1

A

4

A

2

A

3

A

Exchange will depend on the connection graph Actuation: Frequency feedback FACTS: STATCOM, SVC

• Joint work with CISCO Systems and UNC Chapel Hill

SLIDE 34

Phasor Lab

27

SLIDE 35

Sources of Data & Validation

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 Voltage (pu) Bus 1 Bus 2 Midpoint

50 100 150 200 250 300 0.5 1 1.5 2 2.5 3 3.5 Time (sec) Swing Component of Potential Energy ( VA-s ) 50 100 150 200 250 300 0.5 1 1.5 2 2.5 3 3.5 Time(sec) Kinetic Energy ( MW-s )

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 FREEDM facilities

28

SLIDE 36

Conclusions

• 1. WAMS is a tremendously promising technology for smart grid researchers
• 2. Different disciplines must merge
• 3. Plenty of new research problems – EE, Applied Math, Computer Science
• 4. Plenty of new engineering problems
• 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

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SLIDE 37

Thank You

Email: aranya.chakrabortty@ncsu.edu Homepage: people.engr.ncsu.edu/achakra2

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