A Measurement-based Framework for Modeling, Analysis and Control of Large-Scale Power Systems
using Synchrophasors
Aranya ChakraborttyElectrical & Computer Engineering Department
Texas Tech University
North Carolina State University, Raleigh, NC 26th April, 2010
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
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
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
Two-area Model Estimation
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
• 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([)(~112 aEjaEaExV
21 xxxxae
• Voltage magnitude : ,)cos()(2|)(~| 221 aaEEcxVV 2
122
22)1( EaEac
IME: Method (Reactance Extrapolation)
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
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
Reactance Extrapolation
At Bus 1, *)()( 2111, taaAV Busn
• 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
At Bus 2, *)()( 2222, taaAV Busn
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
Reactance Extrapolation
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
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
)()()()( 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
*)()(*)()(*)()(*)()(
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
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
22
),()(
),()(
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 & Sczodrak (2010)
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
23
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
24
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 ?
24
PMU based Disturbance Modeling• Is there an analytical model for disturbance propagation
in power networks?
))1(1()( ...,2,1 ),()}1())1(1{()(1
tptNittptptp
jiji
iiii
• How does mode propagation depend on network shape?
Radial
Star
Cyclic Arbitrary
Well-used Markovian Models in Computer Networks
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
25
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
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
Phasor Lab
27
Phasor Lab
27
Conclusions1. 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
28
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