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Phasor-Only State Estimation
Ali Abur
Department of Electrical and Computer Engineering
Northeastern University, Boston
PSERC Webinar
October 7, 2014
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Talk Outline
Background and terminology
State estimation
Phasor measurements
Phasor-only state estimationWLS: exact cancellations
LAV: improved robustness
Extension to 3-phase networks
Observability and error identificationClosing remarks
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Background and terminology
(Static) state estimation
Measurement type: SCADA / Synchrophasor
Formulation: Nonlinear / Linear
Network model: + sequence / 3-phase
Dynamic state estimation
Wide-area, using gen/load/network variables
Local, using gen variables, boundary measurementsor estimates
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Modeling
Gen
1
Gen
2
Gen
NG
Load 1 Load 2
Load NLTransmission Network[Lines + Transformers]
Measurements4
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Modeling [ SSE ]
Gen
1
Gen
2
Gen
NG
Load 1 Load 2
Load NL
Transmission Network
[Lines + Transformers]
Estimated Measurements
SSE: Estimation of the snap-shot
at time t
Linear or NL
Pos Seq or 3-Phase
SCADA or Phasor or
Mixed SCADA/Phasor
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Modeling [ DSE ]
Gen
1
Gen
2
Gen
NG
Load 1 Load 2
Load NL
Transmission Network
[Lines + Transformers]
Estimated Measurements
DSE: Single machine / zonal / wide-area
Detailed Gen and Load modelsFrequency and power angle estimation
Zone-2
Zone-1
Zone NZ
Tracking the network and
dynamic gen/load state
variables in real-time
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Timeline
[z(t)]
()
()
+ + 2 + 3 + 4
DSE /
Local
SSE /
Wide-area
DSE /
Wide-area
SSE
DSE
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Timeline
[z(t)]
()
()
+ + 2 + 3 + 4
DSE /
Local
SSE /
Wide-area
DSE /
Wide-area
DSE
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Talk Outline
Background and terminology
State estimation
Phasor measurements
Phasor-only state estimationWLS: exact cancellations
LAV: improved robustness
Extension to 3-phase networks
Observability and error identificationClosing remarks
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Static state estimation:
Objective and measurements
Objective: To obtain the best estimate of the state of the system
based on a set of measurements.
State variables: voltage phasors at all buses in the system
Measurements:
Power injection measurements
Power flow measurements
Voltage/Current magnitude measurements
Synchronized phasor measurements
SCADA
Measurem
ents
Introduced by Schweppe and his co-workers in 1969.
Schweppe, F.C., Wildes, J., and Rom, D., Power system static state estimation: Parts I, II, III,Power Industry Computer Applications (PICA), Denver, CO, June 1969.
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State estimation: data/info flow diagram
Topology Processor
Network Observability Analysis WLS State Estimator
Bad Data Processor
Parameter and Topology Error Processor
Analog MeasurementsPi, Qi, Pf, Qf, V, I
TopologyProcessor
StateEstimator
NetworkObservability Check
V,Bad DataProcessor
Network Parameters, Branch Status,
Substation Configuration
Parameter and Topology Errors
Detection, Identification,
Correction
Load Forecasts
Generation Schedules
PM +
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Topology Processor
Network Observability Analysis Robust LAV State Estimator
Bad Data Processor
Parameter and Topology Error Processor
Analog MeasurementsPi, Qi, Pf, Qf, V, I
TopologyProcessor
StateEstimator
NetworkObservability Check
V,Bad DataProcessor
Network Parameters, Branch Status,
Substation Configuration
Parameter and Topology Errors
Detection, Identification,
Correction
Load Forecasts
Generation Schedules
State estimation: data/info flow diagram
PM +
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Talk Outline
Background and terminology
State estimation
Phasor measurements
Phasor-only state estimationWLS: exact cancellations
LAV: improved robustness
Extension to 3-phase networks
Observability and error identificationClosing remarks
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Phasor measurement units (PMU)
Synchronized Phasor Measurements and TheirApplications by A.G. Phadke and J.S. Thorp
Invented by Professors Phadke and Thorp in 1988.
They calculate the real-time phasor measurements
synchronized to an absolute time reference provided by
the Global Positioning System.
PMUs facilitate direct measurement of phase angle
differences between remote bus voltages in a power grid.
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Phasor Measurement Units (PMU)Phasor Data Concentrators (PDC)[*]
GPS CLOCK
AntennaPMU 1
PT
CT
PDC
[*] IEEE PSRC Working Group C37 Report
CORPORATE
PDC
REGIONAL
PDC
DATA STORAGE
& APPLICATIONS
DATA STORAGE
& APPLICATIONS
PMU 2
PMU 3
PMU 4PMU K
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Taken verbatim from: IEEE C37.244-2013 PDC Guide
PDC Definition:
A function that collects phasor data, and discrete event
data from PMUs and possibly from other PDCs, andtransmits data to other applications.
PDCs may buffer data for a short time period, but do not
store the data. This guide defines a PDC as a function
that may exist within any given device.
Phasor Data Concentrator (PDC)
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Phasor voltage measurements
Reference
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Reference phasor
+1
--2
--3 +1
--2
--3
Reference
Reference
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US Energy Information Administration: http://www.eia.gov/todayinenergy/detail.cfm?id=5630
Under the Recovery Act SGIG and SGDP programs, their numbers rapidly increased : 166 1126
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Measurements provided by PMUs
PMU
V phasor
I phasors
ALL 3-PHASES ARE TYPICALLY MEASURED
BUT
ONLY POSITIVE SEQUENCE COMPONENTS ARE REPORTED
= []0+
+ =
[+ +
]
=
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Talk Outline
Background and terminology
State estimation
Phasor measurements
Phasor-only state estimationWLS: exact cancellations
LAV: improved robustness
Extension to 3-phase networks
Observability and error identificationClosing remarks
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Measurement equations
onlyparametersnetworkofFunctionH
ModelLinearXHZ
tsMeasuremenPhasor
XhH
ModellinearNonXhZtsMeasuremenSCADA
x
:
)(:
)(
+=
+=
A.G. Phadke, J.S. Thorp, and K.J. Karimi, State Estimation with Phasor Measurements,
IEEE Transactions on Power Systems, vol. 1, no.1, pp. 233-241, February 1986.
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Talk Outline
Background and terminology
State estimation
Phasor measurements
Phasor-only state estimationWLS: exact cancellations
LAV: improved robustness
Extension to 3-phase networks
Observability and error identificationClosing remarks
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Phasor-only WLS state estimation
varianceerroriiR
ERHRHG
solutionDirectZRHGX
sidualrXHZrtoSubject
rMinimize
problemestimationstateWLS
ModelLinearXHZ
i
TT
T
m
i i
i
),(:
)cov(}{;
e
:
2
1
11
2
2
===
=
=
+=
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Phasor-only WLS state estimation
[ ]
matrixincidencebus-branch:
matrixadmittancebranch:
matrixidentity:
A
Y
U
VAY
U
currentsBranch
voltagesBus
I
VZ
b
b
m
m
m
+
=
=
Consider a fully measured system:
Note: Shunt branches are neglected initially, they will be introduced later. 25
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Phasor-only WLS state estimation
[ ]
[ ] [ ]
++=
++
+=
+
+=
jfeH
jfeAjbg
U
jdc
jfe
I
VLet
F
mm
mm
m
m
)(
26
Ph l WLS i i
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Phasor-only WLS state estimation:
Complex to real transformation
[ ]
+
=
+
=
feHZ
f
e
gAbA
bAgA
U
U
d
c
f
e
m
m
m
m
][
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Ph l WLS t t ti ti
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Phasor-only WLS state estimation:
Correction for shunt terms
XHRHGX
XHZRHGX
ZRHGXXE
XEHuE
XHuuXHXHHZ
shT
sh
Tcorr
T
sh
sh
sh
)(
}{
}{}{
)(][
11
11
11
=
=
==
=
+=+=++=
Very sparse
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System Case
MSE
WLSDecoupled
WLS
A1 0.59 0.592 0.59 0.59
3 0.62 0.62
B
1 0.61 0.61
2 0.61 0.603 0.62 0.63
C
1 0.59 0.59
2 0.58 0.59
3 0.58 0.59
Average MSE Values ( for 100 Simulations)
Fast Decoupled WLS Implementation Results
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MEAN CPU TIMES OF 100 SIMULATIONS
System Case
CPU Times (ms)
WLSDecoupled
WLS
A1 5 2.42 5.7 2.7
3 9.3 3.9
B
1 7.5 3.5
2 8.7 3.9
3 14.8 5.8
C
1 137.4 75.9
2 169.5 95.7
3 284.7 165.6
Fast Decoupled WLS Implementation Results
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Talk Outline
Background and terminology
State estimation
Phasor measurements
Phasor-only state estimationWLS: exact cancellations
LAV: improved robustness
Extension to 3-phase networks
Observability and error identificationClosing remarks
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L1 (LAV) Estimator
]1,,1,1[
||1
=
+=
=
T
m
i
T
c
rXHZtoSubject
rcMinimize
Robust but with known deficiency:
Vulnerable to leverage measurements
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No leverage points
L1 estimator successfully rejects bad data
Motivating example: simple regression
Wrong data
x
yy
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Leverage point exists and contains gross error:
L1 estimator fails to reject bad measurement
Wrong data
(leverage point)
Motivating example: simple regression
y
x
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Leverage points in measurement model
Flow measurements on the lines with impedances, which arevery different from the rest of the lines.
Using a very large weight for a specific measurement.
Mili L., Cheniae M.G., and Rousseeuw P.J., Robust State Estimation of Electric Power
Systems IEEE Transactions on Circuits and Systems, Vol. 41, No. 5, May 1994, pp.349-358.
1
=
xxx
YYY
xxx
xxx
xxx
xxx
HR
LEVERAGE
MEASUREMENT
zexH =+
Scaling both the measured value and the measurement jacobian row will
eliminate leveraging effect of the measurement.
Leave zero injections since they are error free by design.
Incorporate equality constraints in the formulation.37
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Properties of L1 estimator
Efficient Linear Programming (LP) code existsto solve it for large scale systems.
Use of simple scaling eliminates leverage
points. This is possible due to the type ofphasor measurements (either voltages orbranch currents).
L1 estimator automatically rejects bad datagiven sufficient local redundancy, hence baddata processing is built-in.
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Conversion to Equivalent LP Problem
zrHxts
rT
=+..
||cmin
0
..cmin
=
y
zMyts
yT
][][
]00[c
IIHHMVUXXy
cc
TTTT
b
T
a
mmnn
T
==
=
VUrXXx ba
== -
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d d
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Bad data processing
LP problem is solved once. Choice of initial basis impactssolution time/iterations.
Update the
estimates
Normalized Residuals Test (NRT):
NRT is repeated as many times as the number of bad data.
WLS: Post estimation bad data processing / re-estimation
L1: Linear Programming Solution
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Small system example
IEEE 30-bus system Line 1-2 parameter is
changed to transform 1into a leverage measurement
Case 1: Base case true solution
Case 2: Bad leverage
measurement without scaling
Case 3: Same as Case 2, but using
scaling
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Small system example
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Case-1 Case-2 Case-3
V (pu)
(deg) V (pu)
(deg) V (pu)
(deg)
1 1 5.2 1.0118 -3.5462 1 5.2
2 1 -3.74 0.9999 -3.74 1 -3.74
3 0.9952 0.13 1.004 -3.8964 0.9952 0.13
4 0.9992 -4.81 0.9992 -4.81 0.9992 -4.81
5 1 -9.02 0.9999 -9.02 1 -9.02
6 0.9851 -7.41 0.9851 -7.41 0.9851 -7.41
7 0.9869 -8.88 0.9869 -8.88 0.9869 -8.88
Small system exampleBase case
True Solution Leverage Measurements
Bad data
No scalingLeverage Measurements
Bad data
With scaling
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Large test case:
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Large test case:3625 bus + 4836 branch utility system
Case a: No bad measurement.
Case b: Single bad measurement.
Case c: Five bad measurements.
Case a Case b Case c
LAV 3.33 s. 3.36 s. 3.57 s.
WLS 2.32 s. 9.38 s. 50.2 s.
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Phasor only state estimation:
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Phasor-only state estimation:
WLS versus L1 (LAV)
WLS : Linear solution
Requires bad-data analysis
Normalized residuals test
Re-weighting (not applicable) No deficiency in the presence of
leverage measurements, with scaling.
Exact cancellations in [G]
L1 (LAV): Linear programming (single solution)
computationally competitive in s-s
Does not require bad-data analysis[*]
No deficiency in the presence of leverage
measurements, with scaling.
[*] Kotiuga W. W. and Vidyasagar M., Bad Data Rejection Properties of WeightedLeast Absolute Value Techniques Applied to Static State Estimation, IEEE
Transactions on Power Apparatus and Systems, Vol. PAS-101, No. 4, April 1982,
pp. 844-851.
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Talk Outline
Background and terminologyState estimation
Phasor measurements
Phasor-only state estimationWLS: exact cancellations
LAV: improved robustness
Extension to 3-phase networks
Observability and error identificationClosing remarks
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Further motivation
Develop a state estimator capable of solvingstate estimation for three-phase unbalanced
operating conditions,
yet:
Utilize existing, well developed and tested
software as much as possible.
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Proposed approach:
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Proposed approach:
Modal decomposition revisited
Decompose phasor measurements into theirsymmetrical components
Estimate individual symmetrical components
of states separately ( in parallel if such
hardware is available )
Transform the estimates into phase domain to
obtain estimated phase voltages and flows.
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Modal decomposition of
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Modal decomposition ofmeasurement equations
eHVZ +=
][ TIT
V
T ZZZ =
PS TVV = PS TII =
=
T
T
T
TZ
000
0
00
eTHVTZT ZZZ +=
Phasor domain measurement representationH: 3mx3n
V: 3nx1
Z: 30x1
&Modal domain vectors
T: 3x3
VS and IS: 3x1
T: 3mx3m
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Modal decomposition of
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=
1
11
00
00
00
T
TT
TV
MVVTV=
MMMM eVHZ +=
eTe
HTTH
ZTZ
ZM
VZM
ZM
===
0000 eVHZ += rrrr eVHZ +=
0
1
00
1
00 ZRHGV T =
rr
T
rrr ZRHGV 11 =
01
000 HRHG T =
Zero sequence Positive/ Negative sequence
Z0, Zr : mx1
V0, Vr: nx1
H0, Hr: mxn!
rrTrr HRHG
1=
Fully decoupled three relations
Smaller size (Jacobian) matrices
Modal decomposition ofmeasurement equations
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Large scale 3-phase system example:
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Large scale 3 phase system example:
3625-buses and 4836-branches
Case a Case b Case c
LAV WLS LAV WLS LAV WLS
Zero Seq. 3.52 s. 2.32 s. 3.65 s. 9.28 s. 3.78 s. 25.51 s.
Positive Seq. 3.61 s. 2.62 s. 3.63 s. 9.41 s. 3.66 s. 26.01 s.
Negative Seq. 3.21 s. 2.22 s. 3.45 s. 9.01 s. 3.62 s. 24.82 s.
Case a: No bad measurement.
Case b: Single bad measurement.
Case c: Five bad measurements.
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No reference bus or reference PMU
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No reference bus or reference PMUis needed or should be used
Eliminate the reference phase angle from the
SE formulation.
Bad data in SCADA as well as phasor
measurements can be detected and identified
with sufficiently redundant measurement sets.
PMU
Meas.
SCADA
Meas.
State
Estimator &
Bad Data
Processor
Estimated/Corrected
PMU&SCADA Meas.
Estimated State
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Numerical Example
: Power Injection : Power Flow : Voltage Magnitude
: PMU54
Numerical Example
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Numerical ExampleError in bus 1 phase angle
TEST A
Bus 1 is used as the reference bus
TEST B
No reference is used
Measurement
Normalized
residual Measurement
Normalized
residual
7.5 10.53
5.73 7.2
5.6 5.48
5.2 5.35
4.53 4.96
Test A Test B
8 1
74p 8
65p 74p
12
65p
1V 12
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M i Ob bl I l d
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Merging Observable Islands
PMUs can be placed at any bus in theobservable island.
SCADA pseudo-measurements can merge
observable islands only if they are incident tothe boundary buses.
P PS
S
56
R b t M t i
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Robust Metering
Bad data appearing in critical measurementscan NOT be detected.
Adding new measurements at strategic locationswill transform them, allowing detection of bad
data which would otherwise have been missed.
Note:
When a critical measurement is removed fromthe measurement set, the network will no longer beOBSERVABLE.
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Number of Critical Meas Number of PMU needed
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IEEE 57-bus systemCritical
Meas.
Type
1 F41-43
2 F36-35
3 F42-41
4 F40-56
5 I-11
6 I-24
7 I-39
8 I-37
9 I-46
10 I-48
11 I-56
12 I-57
13 I-34
P
P
Number of Critical Meas. Number of PMU needed
13 2
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IEEE 118 b s s stemNumber of Critical Meas. Number of PMU needed
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IEEE 118-bus system29 13
P
P
P
P
P
P
P
P
P
P
P
P
P
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Impact of PMUs on
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p
Parameter Error Identification
Are there cases where errors in certain
parameters can not be identified without
synchronized phasor measurements?
Cases where more than one set of parameters
satisfies all SCADA measurements.
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Illustrative Example
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Illustrative Example
),,(),,( 2121 ppxJppxJ =
k lkl
klP x
=
lm
mllm
xP
=
' 'kl k l
kl k l
xx
=
' 'lm l m
lm l m
xx
=
l mlm
lm
Px
=
kl
lkkl
xP
=
Pl=Plm - Pkl
VkVl
Vm61
Illustrative Example
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Illustrative Example
),,(),,( 2121 ppxJppxJ =
k lkl
klP x
=
lm
mllm
xP
=
' 'kl k l
kl k l
xx
=
' 'lm l m
lm l m
xx
=
l mlm
lm
Px
=
kl
lkkl
xP
=
Pl=Plm - Pkl
Vk
Vl
Vm62
Remarks and Conclusions
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Remarks and Conclusions
LAV estimator will be a computationally viable and effective
alternative to WLS estimator when the measurement set
consists of only PMUs. Scaling can be a simple yet effective
tool in the presence of leverage measurements.
Use of phasor measurements for SE allows modaldecomposition of measurement equations.
WLS estimator implementation has built-in simplifications due
to exact cancellations in the gain [G] matrix.
Given sufficiently redundant phasor measurements, fast androbust state estimation is possible even for very large scale
power grids.
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Related Publications:
Gl, M. and Abur, A., A Hybrid State Estimator for Systems with Limited Number of PMUs, accepted for
publication in IEEE Transactions on Power Systems, 2014.
Gl, M. and Abur, A., LAV Based Robust State Estimation for Systems Measured by PMUs, IEEE Transactions
on Smart Grid, Vol. 5, No: 4, July 2014, pp.1808 -- 1814.
Gl, M. and Abur, A., A Robust PMU Based Three-Phase State Estimator Using Modal Decoupling, IEEE
Transactions on Power Systems, Nov. 2014, Vol. 29, No: 5, pp.2292-2299.
Gl, M. and Abur, A., Observability Analysis of Systems Containing Phasor Measurements, Proceedings of
the IEEE Power and Energy Society General Meeting, 22-26 July 2012.
Gl, M. and Abur, A., Effective Measurement Design for Cyber Security, Proceedings of the 18
th
PowerSystems Computation Conference, August 18-22, 2014, Wroclaw, Poland.
Gl, M. and Abur, A., PMU Placement for Robust State Estimation, Proceedings of the North American
Power Symposium, Sep. 22-24, 2013, Manhattan, KS.
Acknowledgements
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g
National Science Foundation NSF/PSERC
DOE/Entergy Smart Grid Initiative Grants
NSF/ERC CURENT
Jun Zhu, Ph.D. 2009
Jian Chen, Ph.D. 2008
Murat Gol , Ph.D. 2014
Former PhD Students:
Liuxi Zhang, Ph.D. 2014
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Thank You
Any Questions?
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