Seminario (via Web) “Use of Synchronized Phasors in State...

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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)

Transcript of Seminario (via Web) “Use of Synchronized Phasors in State...

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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)

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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

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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.

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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

+==

)()(

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Impact on Problem Solution:

iterativeNonZRHRHX

eXHZtsMeasuremenPhasor

IterativeZRHRHX

eXhZtsMeasuremenalConvention

T

T

−=

+⋅=

∆=∆

+=

−−−

−−−

111

111

)(ˆ

)(ˆ)(

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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.

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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

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Numerical Example

: Power Injection : Power Flow : Voltage Magnitude

: PMU 11 (c) 2013 Ali Abur

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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

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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.

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PMU Placement Problem

matrix incidence Bus [A]

bus for cost onInstallati

=

=

=

iiwotherwise

ibusatinstalledisPMUaifix

01

1̂tosubject

minimize

∑ ⋅

AX

n

i ixiw

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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.

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Simple Illustration of PMU Placement for Full Observability

Only 7 PMUs make the entire system observable.

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Simple Illustration of PMU Placement for Full Observability

Only 7 PMUs make the entire system observable.

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Simple Illustration of PMU Placement for Full Observability

Only 7 PMUs make the entire system observable.

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Simple Illustration of PMU Placement for Full Observability

Only 7 PMUs make the entire system observable.

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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

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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

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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

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Presenter
Presentation Notes
Here is a example. For the given five-bus system, after the transformation, we get such a transformed lower factor. It is notice that there two specific columns in this matrix, the first column and fourth column, each column has only one non-zero elements. The non-zero elements in these two columns correspond to critical measurements, which is power injection measurements in bus 2 and power flow measurement from bus1 to bus2.
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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

56 (c) 2013 Ali Abur

Presenter
Presentation Notes
Here is a example. For the given five-bus system, after the transformation, we get such a transformed lower factor. It is notice that there two specific columns in this matrix, the first column and fourth column, each column has only one non-zero elements. The non-zero elements in these two columns correspond to critical measurements, which is power injection measurements in bus 2 and power flow measurement from bus1 to bus2.
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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

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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:

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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

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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.

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Approach Use of Multiple Scans

][)],,,,([][

][)],,,,([][

][)],,,,([][

21

2222

21

2

1112

11

1

kkn

kkk

n

n

epxxxhz

epxxxhzepxxxhz

+=

+=

+=

][])][],([[][ EpYhZ +=

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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

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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

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• 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].

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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

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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

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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%.

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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.

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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

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