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Ali Abur Department of Electrical and Computer Engineering
Northeastern University
Miércoles 8 de mayo de 2013, de 18:00 a 20:00 horas, en Aula 201 de la ETS de Ingeniería, Universidad de Sevilla
“Use of Synchronized Phasors in State Estimation and its Satellite Functions”
Seminario (via Web)
State Estimation
Analog Measurements Pi , Qi, Pf , Qf , V, I, θkm
Circuit Breaker Status
State Estimator
Bad Data Processor
Network Observability Check
Topology Processor
V, θ
Assumed or Monitored
Pseudo Measurements
2 (c) 2013 Ali Abur
Measurement Model Given a set of measurements, [z] and the correct network topology/parameters: [z] = [h ([x]) ] + [e] Measurements Errors System
State
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SCADA Measurements
Analog: • Power Flow (MW, MVAR) • Power Injection (MW, MVAR) • Voltage Magnitude (V) • Current Magnitude (Amps) • Taps Status: • Breakers • Switches
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Global Positioning Satellites GPS
REFERENCE
V
θ
Phasor Measurements
θ
V
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PMU Measurements: Several (3 phase ) channels measuring:
Phase angles or magnitudes of bus voltages Phase angles or magnitudes of branch currents
Contains a receiver which uses the signals from the global positioning satellite (GPS) system to time stamp samples.
These are processed and converted to phasors 30 times a second.
6 (c) 2013 Ali Abur
State Estimation with PMUs Mathematical model
r
xhxh
xhxhxh
zz
zzz
I
I
V
trad
I
I
V
trad
+
=
)()(
)()()(
)Im(
)Re(
)Im(
)Re(
θθ
voltage magnitudemeasurements
voltage anglemeasurements
current real part measurements
current imaginary part measurements
rxhztoSubjectWrrxJMinimize T
+==
)()(
7 (c) 2013 Ali Abur
Impact on Problem Solution:
iterativeNonZRHRHX
eXHZtsMeasuremenPhasor
IterativeZRHRHX
eXhZtsMeasuremenalConvention
T
T
−=
+⋅=
∆=∆
+=
−−−
−−−
111
111
)(ˆ
)(ˆ)(
8 (c) 2013 Ali Abur
Choosing a reference PMU If the chosen reference bus has no PMU : Reference angle at the chosen bus and actual phase angle measurements provided by PMUs at other buses will not be consistent. If the chosen reference bus has a PMU and measured phase has an error: Error in reference bus phase angle can not be detected. Worse yet, some of the other measurements will be identified as bad data.
9 (c) 2013 Ali Abur
Removal of Reference Bus Eliminate the reference phase angle from
the SE formulation. Bad data in conventional as well as PMU
measurements can be detected and identified with sufficiently redundant measurement sets.
PMU Meas.
Other Meas.
State
Estimator &
Bad Data
Processor
Estimated/Corrected PMU Meas.
Estimated State
10 (c) 2013 Ali Abur
Numerical Example
: Power Injection : Power Flow : Voltage Magnitude
: PMU 11 (c) 2013 Ali Abur
Numerical Example
MeasurementNormalized
residual MeasurementNormalized
residual
7.5 10.535.73 7.25.6 5.485.2 5.35
4.53 4.96
Test A Test B
8θ 1θ
74−p 8θ
65−p 74−p12θ 65−p1V 12θ
Error in bus 1 phase angle
Bus 1 is used as the reference bus No reference is used
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Network Observability [z] = [h ([x]) ] + [e]
Given [z], can [x] be estimated? Which branch flows can be estimated? Unobservable branches separate
observable islands. How to merge observable islands? Optimal measurement placement. Role of pseudo measurements.
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Merging Observable Islands Using Pseudo-injections
ISLAND 1 ISLAND 2
ISLAND 3
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Unobservable Branches PMU
PMU
PMU
PMU
Merging Observable Islands Using PMUs
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Vulnerable Design
PMU
PMU
PMU
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Less Vulnerable Design
PMU
PMU
PMU
Cur
Cur
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A Practical Advantage
PMUs can be placed at any bus in the observable island.
Pseudo-measurements can merge observable islands only if they are incident to the boundary buses.
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Loss of Observability due to Line Switching
OBSERVABLE CLOSED CB
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Loss of Observability due to Line Switching
UNOBSERVABLE BRANCHES IN RED
OPEN CB
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Robustness Against Topology Changes
Observable Observable
Observable Unobservable
Base Case
Base Case
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Meter Placement Given a set of measurements z = h(x) + e Must be able to estimate x using “z” if
any one measurement is missing, or any branch is disconnected
Accomplish the above with least cost metering upgrade
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Three stage solution
Make the network minimally observable. Identify essential measurements
Find a set of candidate measurements. To fix each contingency ( a branch outage or loss
of a measurement).
Determine the optimal selection. To provide secure state estimation under single
contingencies.
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Stage 1: Identify the essential measurements
Conduct Network Observability Analysis Use the Topological Approach Find a set of measurements such that their assigned branches form a spanning tree This set will be called the essential measurements It will not be unique
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Stage 2: Selecting Candidates
=
r
e
HH
Hn essential measurements
(m - n) redundant measurements
[ ]er
e UML
H
=
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Loss of a measurement
0 L
M CANDIDATES
=
r
rede
HH
H modn-1
m- n
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Loss of a branch
= mod
modmod
r
e
HHH
n
m- n
0 L
M CANDIDATES
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Candidate Matrix [A] CANDIDATES
Contingencies
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Stage 3: Optimal Selection
XCT ⋅minimize
bXA ≥⋅Subject to
0/1 Integer Programming
=01
ijA If meas. j is a candidate for contingency i otherwise
( )
=01
iXIf meas. i is selected otherwise
bT=[1 1 1 … 1] ( )
=0ic
iCcost of installing meas. i if meas. i already exist
29 (c) 2013 Ali Abur
30-Bus System Example
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PMU Placement for Full Network Observability
[zPMU] = [H] [x] + [e]
Select the PMU locations to make it possible to estimate [x]
Account for constraints on: Number of available PMU channels Substations without required infrastructure
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PMUs with Unlimited Channels:
PMU V phasor
I phasors
PMU will measure: • V_phasor at the bus • I_phasors for all incident branches
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Simple Illustration of PMU Placement for Full Observability
Only 3 PMUs make the entire system observable. Note that bus 7 is a zero injection bus.
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Simple Illustration of PMU Placement for Full Observability
Only 3 PMUs make the entire system observable. Note that bus 7 is a zero injection bus.
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Simple Illustration of PMU Placement for Full Observability
Only 3 PMUs make the entire system observable. Note that bus 7 is a zero injection bus.
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Simple Illustration of PMU Placement for Full Observability
Only 3 PMUs make the entire system observable. Note that bus 7 is a zero injection bus.
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Optimal PMU Placement
Set up an integer programming problem where all the system buses are made observable by placing PMUs at a minimum number of buses throughout the system.
If there are PMUs already installed or planned to be
installed at specific buses, start with these already pre-assigned to their respective buses.
If there are zero injection buses, incorporate this
information into the placement logic and save as many PMUs as possible.
37 (c) 2013 Ali Abur
PMU Placement Problem
matrix incidence Bus [A]
bus for cost onInstallati
=
=
=
iiwotherwise
ibusatinstalledisPMUaifix
01
1̂tosubject
minimize
≥
∑ ⋅
AX
n
i ixiw
38 (c) 2013 Ali Abur
Simulations (With and without accounting
for zero injection buses)
Systems No. of zero injections
Number of PMUs
Ignoring zero Injections
Using zero injections
14-bus 1 4 3
57-bus 15 17 12
118-bus 10 32 29
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Branch PMUs:
PMU V phasor
I phasor
Assumed configuration for a PMU with two channels: V_phasor at the bus I_phasor along a single branch
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Simple Illustration of PMU Placement for Full Observability
Only 7 PMUs make the entire system observable.
41 (c) 2013 Ali Abur
Simple Illustration of PMU Placement for Full Observability
Only 7 PMUs make the entire system observable.
42 (c) 2013 Ali Abur
Simple Illustration of PMU Placement for Full Observability
Only 7 PMUs make the entire system observable.
43 (c) 2013 Ali Abur
Simple Illustration of PMU Placement for Full Observability
Only 7 PMUs make the entire system observable.
44 (c) 2013 Ali Abur
Simple Illustration of PMU Placement for Full Observability
Only 7 PMUs make the entire system observable.
45 (c) 2013 Ali Abur
Simple Illustration of PMU Placement for Full Observability
Only 7 PMUs make the entire system observable.
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Impact of Channel Limits
How does the optimal placement change as a function of available number of channels?
Assume that one channel corresponds to one positive sequence measurement.
Number of neighbors of a bus usually has a small upper limit for typical power systems due to sparse interconnections.
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3 Channels
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14-bus example
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Problem Formulation
1̂..
min
≥YBts
YCT
T
]111[ =TC
=
11000100
0011010001011000011100010011000101100001110010011
B
Bus 1
Bus 2
Bus 14
50 (c) 2013 Ali Abur
Simulation Results
10 6 7 10 5 7 10 4 7 10 3 9 11 2 14 15 1
5 30-bus
4 4 5 4 4 4 4 4 3 5 5 2 7 7 1
1 14-bus
Consider Zero Injections
Ignore Zero Injections Channels Zero Injections System
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28 32 9 28 32 8 28 32 7 28 32 6 28 32 5 28 32 4 31 33 3 39 42 2 57 61 1
10 118-bus
11 8
11 7
11 17 6 11 17 5 11 17 4 12 17 3 14 19 2 23 29 1
15 57-bus
Consider Zero
Injections Ignore Zero
Injections Channels Zero Injections System
Simulation Results
52 (c) 2013 Ali Abur
Robustness: How can PMUs help to improve bad data detection?
A measurement is said to be “critical” if the system becomes unobservable upon its removal.
Bad data appearing in critical measurements can NOT be detected.
Adding new measurements at strategic locations will transform them, allowing detection of bad data which would otherwise have been missed.
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Example: 5-bus system with 2 critical measurements
−
=
−
05.05.0005.05.001000010000100001
23
34
12
24
3
2
)1(
PPPPPP
KI
R
n
Null columns in sub-matrix KR indicates the critical
measurements
=⋅
=
−−
R
n
KI
HH
HH
H
111
2
1
54 (c) 2013 Ali Abur
Transforming critical measurements using a single PMU
−
=
−
05.05.0005.05.001000010000100001
23
34
12
24
3
2
)1(
PPPPPP
KI
R
n
1 bus at PMU
25.15.0110000000
15
12
1
IIθ
New measurements make P2 and P12 redundant
P
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PMU Placement: Before and After
P
Two Critical Measurements
NO Critical Measurements
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IEEE 57-bus system Critical 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
57 (c) 2013 Ali Abur
IEEE 118-bus system Number of Critical Meas. Number of PMU needed
29 13
P
P
P
P
P
P
P
P
P
P
P
P
P
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Robustness Against Parameter Errors
Total number of network parameters is typically a very large number
Parameter error processing is not an on-line feature in commonly used estimators
It is usually difficult to pinpoint the right suspect set simply based on residual analysis due to error masking.
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Parameter Error Identification Problem
Identify incorrect parameters associated with power network models.
Decision: Bad data or parameter error? How to make the estimator robust against
bad data and parameter errors?
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Using SCADA Measurements
Existing practice: Augment the state vector with the suspected set of parameters and estimate suspect parameters
Recently proposed improved solution: Eliminates the need to pre-specify a suspect set. Detects and identifies parameter and analog measurement errors simultaneously
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Problem Formulation
et ppp +=
Every network parameter is assumed to have an error:
Parameter Used By the State Estimator
True Value of Parameter
Parameter Error
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Problem Formulation
( ) et
ett ppxcWrrL λµ −−= ,
21
State Estimation with Equality Constraints:
0),(0..
)],([)],([),(min 21
==
−−=
e
e
eT
ee
pxcpts
pxhzWpxhzpxJ
Lagrangian:
63 (c) 2013 Ali Abur
Formulation
∆=
∆
czr
x
G
CIH
CWH TT 0
000
0
µ
=
µλ
rS
T
p
p
CWH
S
−=
First order optimality conditions:
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30-bus system
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Numerical Example: Test I: reactance of line 5-7 is incorrect. Test II: real power flow on line 5-7 is incorrect.
Measurement/ Measurement/
Parameter Parameter
25.47 19.522.01 12.3421.92 10.5615.78 9.9715.42 9.86
Test I Test II
Normalized residual / λN
Normalized residual / λN
75−x 75−p
67−x 75−r
52−x 5p
67−r 6q
52−r 67−x
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Case of Multiple Errors
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Numerical Example: Errors: reactance of line 2-4 transformer tap of TR 4-9 power flow measurement on line 4-2
z/ p r N / λN z/ p r N / λN z/ p r N / λN
60.56 23.87 5.0746.48 17.99 3.7540.49 10 3.0230.24 9.78 2.86
25 9.68 2.25
Identified and Eliminated error
Error identification cycle
1st 2nd 3rd
42−x 94−t 24−p
24−p 49−p 3p
54−x 74−t 4p
52−x 97−r 42−r
94−t 4p 54−p
42−x 94−t 24−p
68 (c) 2013 Ali Abur
Error Correction: Sequential vs. Simultaneous
Bad TRUEParameter Parameter
1st 0.174 0.17632
2nd 0.96015 0.96
StepEstimated Parameter
42−x
94−t
Bad TRUEParameter Parameter
0.17633 0.176320.96 0.96
Estimated Parameter
42−x
94−t
SEQUENTIAL CORRECTION
SIMULTANEOUS CORRECTION
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Limitation: unidentifiable errors
Test : Susceptance of shunt cap at bus 24 is incorrect.
Measurement/
Parameter
12.7212.725.785.234.65
Normalized residual / λN
24b
24q
2422−q
22q
2423−q
Equal
Q
b
24
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How to overcome this limitation?
In certain cases, even with highly redundant measurement sets, the method may fail to identify certain types of errors.
To remove this limitation, integrate multiple measurement scans and reformulate the identification problem.
71 (c) 2013 Ali Abur
Approach Use of Multiple Scans
][)],,,,([][
][)],,,,([][
][)],,,,([][
21
2222
21
2
1112
11
1
kkn
kkk
n
n
epxxxhz
epxxxhzepxxxhz
+=
+=
+=
][])][],([[][ EpYhZ +=
72 (c) 2013 Ali Abur
Minimize
Approach Problem Formulation
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Approach Problem Formulation
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Existing system with network parameters
State estimation
Multiple measurement
scans
Calculation of system states (xi), residuals (ri) and Lagrange multipliers λ
Computation of normalized residuals
rN
Computation of normalized Lagrange
multipliers λN
Compare and identify parameter or measurement
errors
Algorithm
75 (c) 2013 Ali Abur
Test A (Single Scan) Test B (Multiple Scans) Parameter/
Measurement Parameter/ Measurement
7.6305 16.7925 7.6305 7.6349 2.3828 7.6305
Numerical Results
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Numerical Results
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Test A (Single Scan) Test B (Multiple Scans)
Parameter/ Measurement Parameter/
Measurement
7.7012 15.9898
7.7012 7.8153 2.7821 7.7012
Numerical Results
78 (c) 2013 Ali Abur
• Implicit improvement in redundancy without actually adding any new meters. This allows certain types of parameter errors, which cannot be estimated before via single scan, become identifiable.
• It can be implemented without modifying the existing state estimation programs. These programs can be executed repeatedly at different times.
Approach Advantages
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Multiple Solutions
What happens when more than one set of parameters satisfies all measurements? Multiplicity of solutions for [p].
80 (c) 2013 Ali Abur
Illustration of Multiple Solutions
),,(),,( 2121 ppxJppxJ ′′′=
k lkl
kl
Px
θ θ−=
lm
mllm x
P′−′
=θθ
' 'kl k l
kl k l
xx
θ θθ θ−
=−
' 'lm l m
lm l m
xx
θ θθ θ−
=−
l mlm
lm
Px
θ θ−=
kl
lkkl x
P′
′−=
θθ
81 (c) 2013 Ali Abur
Identifying Parameter Errors by Phasor Measurements
: Parameter Error : PMU
TEST A: NO PMUs TEST B: WITH PMUs
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Simulation Results Test A Test B
Meas /Par. RN / λN Meas /Par. RN / λN
X10-11 27.5040 X13-14 29.0162
X13-14 27.3629 X9-14 26.0467
X6-11 27.9763 X10-11 20.1385
X9-14 24.4732 X6-11 20.1349
X9-10 20.2291 X9-10 17.0387
83 (c) 2013 Ali Abur
Synchro Phasor Assisted State Estimator [ SPASE ]
network
measure
Functions: • Network Observability • Pseudo-measurements
Placement • Critical Measurements
Identification • State Estimation • Bad Data Processing • Parameter Error ID
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Output Data Estimated state variables Estimated measurements Critical measurements (MSV metric) Pseudo-measurements (PMR metric) Observable islands Unobservable branches Performance metrics
Objective function (weighted sum of squares of residuals)
Chi-2 threshold for detecting bad data Largest normalized residual for identifying bad data Diagonal of the inverse of gain matrix (SEA metric)
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Metrics
SPASE Performance Measurement Quality Measurement Design
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SPASE Performance
Convergence rate Number of iterations to converge for a given convergence tolerance, e.g. 0.001 per unit. Smaller values indicate better performance ITMAX = Number of iterations to converge ITMAX is independent of system size but dependent on network model and measurement configuration. Expected value is less than 8-10.
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Measurement Quality
Performance Index Assesses the quality of measurements. Expected value is given by Chi-2 distribution and will be used for bad data detection. Threshold = Chi-squares value looked up from the Statistical Table corresponding to (m-n) degrees of freedom, 95% confidence.
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Measurement Quality
Largest Normalized Residual
Checked against a threshold based on Normal and a chosen confidence level. If larger, then the corresponding measurement is flagged as BD.
Threshold is typically chosen as 3.0
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Measurement Design Measurement System Vulnerability (MSV) Metric: Large number of critical measurements indicate vulnerability to bad data. Their locations reveal vulnerability zones and also provide clues for areas to invest for new meters.
MSV = 𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜 𝑐𝑁𝑐𝑐𝑐𝑐𝑐𝑐 𝑁𝑁𝑐𝑚𝑁𝑁𝑁𝑁𝑁𝑚𝑐𝑚𝑇𝑜𝑐𝑐𝑐 𝑚𝑁𝑁𝑁𝑁𝑁 𝑜𝑜 𝑁𝑁𝑐𝑚𝑁𝑁𝑁𝑁𝑁𝑚𝑐𝑚
MSV can be defined with respect to: • Geographical areas (zones) • Voltage levels A robust measurement system is recommended to have an MSV < 3%.
90 (c) 2013 Ali Abur
Measurement Design Pseudo-measurement ratio (PMR): PM are not to be trusted and locations provide information about zones of low redundancy. They also should remain critical to avoid spreading of their errors.
PMR = 𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜 𝑝𝑚𝑁𝑁𝑝𝑜−𝑁𝑁𝑐𝑚𝑁𝑁𝑁𝑁𝑁𝑚𝑐𝑚𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜 𝑜𝑁𝑚𝑁𝑁𝑜𝑐𝑁𝑐𝑁 𝑐𝑚𝑐𝑐𝑚𝑝𝑚
Redundant pseudo-measurements increase chances of corrupting actual (good) measurements. PMR should be close to 1.0 for optimal results.
91 (c) 2013 Ali Abur
Measurement Design Max Diagonal of the inverse gain matrix (G-1): Small values imply better accuracy. Changes in this metric is mainly a function of measurement configuration and not measurement values. State Estimation Accuracy (SEA) metric: SEA = max{ 𝑑𝑑𝑑𝑑 𝐺−1 } SEA value should remain below the acceptable variance of errors in estimated states. Typical threshold is in the order of 10-6.
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Final Remarks
PMUs can help to improve SE performance in the following areas: Computational speed and numerical
robustness Merging observable islands Bad data detection and identification Identifying network parameter errors
93 (c) 2013 Ali Abur