Phasor State Estimation Weighting Coefficients for AC and Hybrid Networks with Power Electronic Devices - or - How to Quantify Measurements Weights from PMUs?

Phasor State Estimation Weighting Coefficients

for AC and Hybrid Networks with

Power Electronic Devices

or How to Quantify Measurements Weights from PMUs?

Wei Li (KTH) and Luigi Vanfretti (RPI)

[email protected] [email protected]

1

IEEE PES General Meeting July 20, 2017, Chicago, USA

mailto:[email protected]

mailto:[email protected]

Motivations

Power electronics-based devices (e.g., flexible AC transmission system (FACTS) and

voltage source converter (VSC)-based HVDC links) installations continues increasing

worldwide. Their real-time performance during dynamic responses that need to be

monitored

A large potential to develop suitable SE algorithms and models to monitor their

dynamical behavior. However, most of the so-called dynamic SEs or forecasting-

aided SEs are computationally demanding

We focus on a pseudo-dynamic PMU-only SE that is capable of addressing system

dynamics with low computational demands. And this SE uses WLS algorithm.

WLS SEs use weights to take into account inaccuracies in measurements and

modeling

This work focuses on how to quantify measurement weights for PMU-only SEs,

mainly for the AC network measurements

2

Outline

Part I: Pseudo-dynamic network modeling for PMU based state estimation of

hybrid AC/DC grids

Formulation

Models

Part II: Approaches on how to quantify measurement variance

Simulation on computers

Hardware-in-the-loop test

Real PMU data (telemetry)

3

Formulation I: WLS and measurement model

Weighted least squares (WLS) and the measurement model

Eq.(1)

where is the error vector and is the th row; is the th diagonal

element of the weight matrix.

The error vector contains two parts:

Network model equations , which may contain modeling errors, and thus,

weights based on the confidence on the model’s accuracy are assigned.

Errors between the measurements and their corresponding states . As PMUs

enable to measure system states directly, the errors are for the quantities

such as , and even other user-defined states. For instance,

4

2

1

( )min ,

n m

i i

i

w e

x

h xe

ε

n me ¡ ie i iw i

e

( ) nh x ¡

mε ¡

,l l m x ¡| |, | |, ,V I θ δ

{ { {ˆ

ii iV m

measurementerror state

V V

Formulation I: advantage

Weighted least squares (WLS) and the measurement model

Eq.(1)

The advantage of using Eq. (1) lies in the flexibility of granting different weights to

different network model equations and measurements:

Network Equations: disparate reliabilities of the model’s parameters.

Measurements: different accuracies depending on instrumentation, internal phasor

algorithm, and other variables.

5

2

1

( )min ,

n m

i i

i

w e

x

h xe

ε

Formulation II: Pseudo-dynamic network model

Network models for the static SE cannot fully represent the states’ time-series

trajectory due to the lack of representation of dynamic properties.

Pseudo-dynamic network model leverages the existing body of network model and

include the difference equations that describe the system dynamic properties.

6

Continuous dynamical system

Differential equations

Telemetry acquired discretely

over time intervals Discrete dynamical system

Difference equations

Euler’s full step modification, can be used to formulate h(x), resulting in the

difference equation:

$ 1 1( ) : ( ) ( ) .2

sk k k k

T kh x x x g x g x

Numerically solve differential

equations, i.e., numerical integration.

$ $..

11 ( )2

sk k kk

T x x x x

Generalized form

Model example: STATCOM

7

1

K

T s

refV

V stI

1| | ( | |) | |ref

st st

KI V V I

T T &

$ 1 1( ) : ( ) ( ) .2

sk k k k

T kh x x x g x g x

using

Pseudo-dynamic model :

refV

VsX

stIstI

max

capImax

indI

Capacitive Inductive

( ) :| | | | ref

s stV X I V h x

,

, 1 1

ˆ ˆ( ) : (1 ) | | | |2 2

(1 ) | | | | .2 2

s sst k k

refs s sst k k

T T KI V

T T

T K T T KV I V

T T T

kh x

| |,| |, , ,| |T

st

x V I θ δ I

Aim to control the voltage at

the connected bus.

A linear V-I relation when it

is under steady state

operation conditions.

Static network model:

Model example: case study

8

A modified WSCC 3-machine 9-bus system; A STATCOM at Bus 8

A 16.67% load increase (both P and Q) at Bus 8 was applied at t = 2s

The magnitude residual by the static SE up to 0.1783 p.u.

The pseudo-dynamic SE’s maximum residual 1.05*10^(-13) p.u.

Using static model Using pseudo-dynamic model

1 2 3 4 5 6 7 80

0.2

0.4

time

|I|(

p.u

.)

Imag-true

Imag-m

Imag-est

1 2 3 4 5 6 7 80

1

2

3x 10

-16

time

Err

or(

p.u

.)

Imag-residual-error

1 2 3 4 5 6 7 80

0.2

0.4

time

|I|(

p.u

.)

Imag-true

Imag-m

Imag-est

1 2 3 4 5 6 7 80

1

2

3x 10

-16

time

Err

or(

p.u

.)

Imag-residual-error

Model example: case study

9

A modified WSCC 3-machine 9-bus system; A STATCOM at Bus 8

A 16.67% load increase (both P and Q) at Bus 8 was applied at t = 2s

The magnitude residual by the static SE up to 0.1783 p.u.

The pseudo-dynamic SE’s maximum residual 1.05*10^(-13) p.u.

Using static model Using pseudo-dynamic model

1 2 3 4 5 6 7 80.1

0.15

0.2

0.25

time|Is

t|(p

.u.)

|Ist|-true

|Ist|-est

1 2 3 4 5 6 7 80

0.5

1x 10

-13

time

Err

or(

p.u

.)

|Ist|-residual-error

1 2 3 4 5 6 7 80

0.2

0.4

time

|Is

t|(p

.u.)

|Ist|-ture

|Ist|-est

1 2 3 4 5 6 7 80

0.1

0.2

time

Err

or(

p.u

.)


1.95 2 2.05 2.1

0.1

0.2

0.3

time

|Is

t|(p

.u.)

|Ist|-m

|Ist|-est0 2 4 6 80

0.05

0.1

0.15

0.2

time

Err

or(

p.u

.)


Outline

Part I: Pseudo-dynamic network modeling for PMU based state estimation of

hybrid AC/DC grids

Formulation

Models

Part II: Approaches on how to quantify measurement variance

Simulation on computers

Hardware-in-the-loop test

Real PMU data (telemetry)

10

Quantification of measurement weights

How should be computed for different phasor measurements

Three approaches are used here: - simulation, -HIL, and field data analysis.

Different scenarios for each approach are studied.

Impact of measurement noise is analyzed for off-line simulation, and HIL

Impact of combined process and measurement noise is analyzed for field data.

11

2

1i

i

w

i

i | |,| |, ,V I θ δ

For WLS, if the errors are independent

and have normal distributions, weights

for measurements are typically specified

as: , where is the standard

deviation of the measurement i.

Simulation on computers: set-up

12

[1] D. Dotta, J. H. Chow and D. B. Bertagnolli, "A Teaching Tool for Phasor Measurement Estimation," in

IEEE Transactions on Power Systems, vol. 29, no. 4, pp. 1981-1988, July 2014.

signal

generationA teaching tool

for phasor

measurement

estimation [1]

Reference PMU

Sequence

Analyzer

Calculate the

standard deviations

for magnitude and

angle

Calculate the

standard deviations

for magnitude and

angle

Simulink/Matlab

33

/ 0

50*32 Hz

50Hz

3-phase signals generation

Perfectly balanced

Ref PMU PMU

Simulation magnitude (1, 1.59259e-13) (1, 4.44534e-15)

Simulation angle (8.91792e-14, 3.43861e-13) (-2.98428e-13, 0)

0.03 0.035 0.04 0.045 0.05 0.055 0.06-1

-0.5

0

0.5

1

1 1.2 1.4 1.6 1.8 2-0.5

0

0.5

1

1.5

refPMU |V|

refPMU

1 1.2 1.4 1.6 1.8 2-0.2

0

0.2

0.4

0.6

0.8

1

1.2

PMU |V|

PMU

Matlab function:

“fitdist”

Distribution type:

“Normal”

-100 -50 0 50 1000

100

200

300

400

500

histogram PMU |V|

histogram refPMU |V|

pdf PMU |V|

pdf refPMU |V|

-100 -50 0 50 1000

100

200

300

400

500

histogram PMU

histogram refPMU

pdf PMU

pdf refPMU

Simulation on computers: set-up

13

[1] D. Dotta, J. H. Chow and D. B. Bertagnolli, "A Teaching Tool for Phasor Measurement Estimation," in

IEEE Transactions on Power Systems, vol. 29, no. 4, pp. 1981-1988, July 2014.

signal

generationA teaching tool

for phasor

measurement

estimation [1]

Reference PMU

Sequence

Analyzer

Calculate the

standard deviations

for magnitude and

angle

Calculate the

standard deviations

for magnitude and

angle

Simulink/Matlab

33

/ 0

50*32 Hz

50Hz

Perfect 3 phase signals – histogram shows a peak at mean.

Assumption of perfect measurement weights equal to 1.


With different Gaussian noise levels. For instance, 10% variation, 0 gain

0.8 0.9 1 1.1 1.2 1.30

2

4

6

8

10

12

histogram PMU |V|


pdf PMU |V|

pdf refPMU |V|

signal

generation

A teaching tool

for phasor

measurement

estimation [1]

Reference PMU

Sequence

Analyzer

Calculate the

standard deviations

for magnitude and

angle

Calculate the

standard deviations

for magnitude and

angle

Simulink/Matlab

33

/ 0

50*32 Hz

50Hz

Gaussian

noise

Introducing emulated measurement noise

14

Ref PMU PMU

Simulation magnitude (1.00123, 0.0473624) (1.00129, 0.0477469)

Simulation angle (-0.0959179, 2.5998) (-0.0948518, 2.13655)

-10 -5 0 5 100

5

10

15

histogram PMU

histogram refPMU

pdf PMU

pdf refPMU

0.03 0.04 0.05 0.06-1.5

-1

-0.5

0

0.5

1

1.5

21 1.5 2

0.5

1

1.5

refPMU |V|

1 1.5 2-5

0

5

10

refPMU

1 1.5 20.8

1

1.2

PMU |V|

1 1.5 2-5

0

5

PMU

Summary of cases with measurement noise

15

0 0.1 0.2 0.3 0.4 0.51

1.002

1.004

1.006

1.008

of the input noise

o

f th

e ou

tpu

t

Magnitude

refPMU noise

PMU noise

0 0.1 0.2 0.3 0.4 0.5-0.25

-0.2

-0.15

-0.1

-0.05

0

of the input noise

o

f th

e ou

tpu

t

Angle

refPMU noise

PMU noise

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

of the input noise

o

f th

e o

utp

ut

Magnitude

refPMU noise

PMU noise

0 0.1 0.2 0.3 0.4 0.50

2

4

6

of the input noise

o

f th

e o

utp

ut

Angle

refPMU noise

PMU noise

Non-linear relationship.

Under the same input noise, model of instrument has impact on the mean for the

magnitude even if the variance is identical:

Different measurement values for the measurement equations.

Summary of cases with measurement noise

16

0 0.1 0.2 0.3 0.4 0.51

1.002

1.004

1.006

1.008

of the input noise

o

f th

e ou

tpu

t

Magnitude

refPMU noise

PMU noise

0 0.1 0.2 0.3 0.4 0.5-0.25

-0.2

-0.15

-0.1

-0.05

0

of the input noise

o

f th

e ou

tpu

t

Angle

refPMU noise

PMU noise

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

of the input noise

o

f th

e o

utp

ut

Magnitude

refPMU noise

PMU noise

0 0.1 0.2 0.3 0.4 0.50

2

4

6

of the input noise

o

f th

e o

utp

ut

Angle

refPMU noise

PMU noise

Non-linear relationship.

Under the same input noise, model of instrument has impact on the variance for the

angle even if the mean is almost identical:

Different weights are needed for different instrument models.

signal

generation

A teaching tool

for phasor

measurement

estimation [1]

Reference PMU

Sequence

Analyzer

Calculate the

standard deviations

for magnitude and

angle

Calculate the

standard deviations

for magnitude and

angle

Simulink/Matlab

33

/ 0

50*32 Hz

50Hz

3rd

harmonics

Introducing harmonics

17


With harmonics. For instance, 3rd harmonics on three phases with 0.5 gain

Another example, 3rd harmonics on one phase with 0.5 gain

Ref PMU PMU

Simulation magnitude (1, 1.20421e-13) (1, 4.63932e-15)

Simulation angle (3.03539e-14, 2.34453e-12) (-2.984e-13, 6.35529e-16)

Ref PMU PMU

Simulation magnitude (1, 8.76263e-14) (1,1.22697e-14)

Simulation angle (1.4167e-14, 7.35846e-13) (-3.09797e-13, 5.80113e-13)

0.94 0.96 0.98 1 1.02 1.04 1.060

100

200

300

400

histogram PMU |V|


pdf PMU |V|

pdf refPMU |V|

-10 -5 0 5 100

100

200

300

400

500

histogram PMU

histogram refPMU

pdf PMU

pdf refPMU

2.985 2.99 2.995 3 3.005 3.01 3.015 3.02 3.025

-1

-0.5

0

0.5

1

1 1.2 1.4 1.6 1.8 2-0.5

0

0.5

1

1.5

refPMU |V|

refPMU

1 1.2 1.4 1.6 1.8 2-0.2

0

0.2

0.4

0.6

0.8

1

1.2

PMU |V|

PMU

signal

generation

A teaching tool

for phasor

measurement

estimation [1]

Reference PMU

Sequence

Analyzer

Calculate the

standard deviations

for magnitude and

angle

Calculate the

standard deviations

for magnitude and

angle

Simulink/Matlab

33

/ 0

50*32 Hz

50Hz

3rd

harmonics

Introducing harmonics

18

Under perfect condition, harmonics are filtered by the PMUs, which is

expected from design.

signal

generation

A teaching tool

for phasor

measurement

estimation [1]

Reference PMU

Sequence

Analyzer

Calculate the

standard deviations

for magnitude and

angle

Calculate the

standard deviations

for magnitude and

angle

Simulink/Matlab

33

/ 0

50*32 Hz

50Hz

Gaussian

noise

3rd

harmonics

Harmonics + measurement noise

19


3rd harmonics on three phases with 0.5 gain + Gaussian noise with 10% standard deviation

1 1.5 20.8

1

1.2

refPMU |V|

1 1.5 2-5

0

5

10

refPMU

1 1.5 20.8

1

1.2

PMU |V|

1 1.5 2-5

0

5

PMU

Ref PMU PMU



0.8 0.9 1 1.1 1.2 1.30

5

10

15

histogram PMU |V|


pdf PMU |V|

pdf refPMU |V|

-10 -5 0 5 100

5

10

15

histogram PMU

histogram refPMU

pdf PMU

pdf refPMU

0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065-3

-2

-1

0

1

2

3

Summary of harmonics + measurement noise

20

0 0.1 0.2 0.3 0.4 0.50

0.02

0.04

0.06

0.08

0.1

0.12

of the input noise

o

f th

e ou

tpu

t

Magnitude

refPMU noise

refPMU noise and harmonics

PMU noise

PMU noise and harmonics

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

of the input noise

o

f th

e ou

tpu

t

Angle

0 0.1 0.2 0.3 0.4 0.51

1.002

1.004

1.006

1.008

1.01

of the input noise

o

f th

e o

utp

ut

Magnitude

refPMU noise


PMU noise


0 0.1 0.2 0.3 0.4 0.5-0.4

-0.3

-0.2

-0.1

0

of the input noise

o

f th

e o

utp

ut

Angle

Introducing harmonics increases the absolute

mean values for both magnitude and angle.

Summary of harmonics + measurement noise

21

0 0.1 0.2 0.3 0.4 0.50

0.02

0.04

0.06

0.08

0.1

0.12

of the input noise

o

f th

e ou

tpu

t

Magnitude

refPMU noise


PMU noise


0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

of the input noise

o

f th

e ou

tpu

t

Angle

0 0.1 0.2 0.3 0.4 0.51

1.002

1.004

1.006

1.008

1.01

of the input noise

o

f th

e o

utp

ut

Magnitude

refPMU noise


PMU noise


0 0.1 0.2 0.3 0.4 0.5-0.4

-0.3

-0.2

-0.1

0

of the input noise

o

f th

e o

utp

ut

Angle

Introducing harmonics does not have much

effect on the variances for both magnitude

and angle.

signal

generation

1,0.95,1 puA teaching tool

for phasor

measurement

estimation [1]

Reference PMU

Sequence

Analyzer

Calculate the

standard deviations

for magnitude and

angle

Calculate the

standard deviations

for magnitude and

angle

Simulink/Matlab

33

/ 0

50*32 Hz

50Hz

Introducing unbalanced 3Φ

22

0 2 4 6 8 100.9833

0.9833

0.9833

refPMU |V|

0 2 4 6 8 10

-5

0

5

x 10-10

refPMU

2 4 6 8 10-2

0

2

PMU |V|

2 4 6 8 10-2

0

2

PMU


Unbalanced three phases. For instance, with magnitude 1, 0.95, 1 for a, b, c phase, respectively

Ref PMU PMU

Simulation magnitude (0.983333, 1.71606e-13) (0.983333, 8.22388e-15)

Simulation angle (6.63111e-13, 3.27147e-12) (-2.98428e-13,0)

-100 -50 0 50 1000

100

200

300

400

500

histogram PMU |V|


pdf PMU |V|

pdf refPMU |V|

-100 -50 0 50 1000

100

200

300

400

500

histogram PMU

histogram refPMU

pdf PMU

pdf refPMU

0.03 0.04 0.05 0.06-1

-0.5

0

0.5

1

signal

generation


for phasor

measurement

estimation [1]

Reference PMU

Sequence

Analyzer

Calculate the

standard deviations

for magnitude and

angle

Calculate the

standard deviations

for magnitude and

angle

Simulink/Matlab

33

/ 0

50*32 Hz

50Hz

Introducing unbalanced 3Φ

23

Under perfect condition, unbalanced three-phase only affects the magnitude of

PMU output

Unbalanced 3Φ + measurement noise

24

signal

generation


for phasor

measurement

estimation [1]

Reference PMU

Sequence

Analyzer

Calculate the

standard deviations

for magnitude and

angle

Calculate the

standard deviations

for magnitude and

angle

Simulink/Matlab

33

/ 0

50*32 Hz

50Hz

Gaussian

noise

0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065-2

-1

0

1

2

1 1.2 1.4 1.6 1.8 20.5

1

1.5

PMU |V|

1 1.2 1.4 1.6 1.8 2-5

0

5

PMU

1 1.2 1.4 1.6 1.8 20.5

1

1.5

refPMU |V|

1 1.2 1.4 1.6 1.8 2-10

0

10

refPMU


Unbalanced three phases with magnitude 1, 0.95, 1 for a, b, c phase, respectively +

Gaussian noise with 10% standard deviation

Ref PMU PMU



0.8 0.9 1 1.1 1.2 1.30

2

4

6

8

10

12

histogram PMU |V|


pdf PMU |V|

pdf refPMU |V|

-10 -5 0 5 100

2

4

6

8

10

12

14

histogram PMU

histogram refPMU

pdf PMU

pdf refPMU

Summary of unbalanced 3Φ+ meas. noise

25

0 0.1 0.2 0.3 0.4 0.50

0.02

0.04

0.06

0.08

0.1

0.12

of the input noise

o

f th

e ou

tpu

t

Magnitude

refPMU noise

refPMU noise and unbalanced 3

PMU noise

PMU noise and unbalanced 3

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

of the input noise

o

f th

e ou

tpu

t

Angle

0 0.1 0.2 0.3 0.4 0.50.98

0.99

1

1.01

1.02

of the input noise

o

f th

e ou

tpu

t

Magnitude

0 0.1 0.2 0.3 0.4 0.5-0.4

-0.3

-0.2

-0.1

0

of the input noise

o

f th

e ou

tpu

t

Angle

Introducing unbalanced three-phase affects

the absolute mean values for both magnitude

and angle.

Summary of unbalanced 3Φ+ meas. noise

26

0 0.1 0.2 0.3 0.4 0.50

0.02

0.04

0.06

0.08

0.1

0.12

of the input noise

o

f th

e ou

tpu

t

Magnitude

refPMU noise

refPMU noise and unbalanced 3

PMU noise

PMU noise and unbalanced 3

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

of the input noise

o

f th

e ou

tpu

t

Angle

0 0.1 0.2 0.3 0.4 0.50.98

0.99

1

1.01

1.02

of the input noise

o

f th

e ou

tpu

t

Magnitude

0 0.1 0.2 0.3 0.4 0.5-0.4

-0.3

-0.2

-0.1

0

of the input noise

o

f th

e ou

tpu

t

Angle

Introducing unbalanced three-phase does

not have much effect on the variances for

both magnitude and angle.

Hardware-in-the-loop test—set up

27

master console

3-phase signal

generation model

in RT-Lab

signal

generation

RT-Lab

signal

streams

Opal-RT real-time

simulator

3 3 · Relay

· A/D

· Phasor

estiamtor

SEL-421

Protection Relays

and PMU 3 3

Collect data

SEL-PDC-5073

3Read data

locally

PMU connection

tester

3


28

Load and execute

the model in the

real-time simulator

signal

generation

RT-Lab

signal

streams

Opal-RT real-time

simulator

3 3 · Relay

· A/D

· Phasor

estiamtor

SEL-421

Protection Relays

and PMU 3 3

Collect data

SEL-PDC-5073

3Read data

locally

PMU connection

tester

3


29

Send out analog 3-

phase signals from

simulator to PMU

signal

generation

RT-Lab

signal

streams

Opal-RT real-time

simulator

3 3 · Relay

· A/D

· Phasor

estiamtor

SEL-421

Protection Relays

and PMU 3 3

Collect data

SEL-PDC-5073

3Read data

locally

PMU connection

tester

3


30

Read and capture

the PMU streams

from the PDC

signal

generation

RT-Lab

signal

streams

Opal-RT real-time

simulator

3 3 · Relay

· A/D

· Phasor

estiamtor

SEL-421

Protection Relays

and PMU 3 3

Collect data

SEL-PDC-5073

3Read data

locally

PMU connection

tester

3


31

signal

generation

RT-Lab

signal

streams

Opal-RT real-time

simulator

3 3 · Relay

· A/D

· Phasor

estiamtor

SEL-421

Protection Relays

and PMU 3 3

Collect data

SEL-PDC-5073

3Read data

locally

PMU connection

tester

3

Results of HIL testGaussian noise in the measurement input

32

0 0.1 0.2 0.3 0.4 0.50.97

0.98

0.99

1

1.01

1.02

of the input noise

o

f th

e ou

tpu

t

Magnitude

offline noise

HIL noise

0 0.1 0.2 0.3 0.4 0.5-0.3

-0.2

-0.1

0

0.1

of the input noise

o

f th

e ou

tpu

t

Angle

offline noise

HIL noise

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

of the input noise

o

f th

e ou

tpu

t

Magnitude

offline noise

HIL noise

0 0.1 0.2 0.3 0.4 0.50

2

4

6

of the input noise

o

f th

e ou

tpu

t

Angle

offline noise

HIL noise

Results of HIL testGaussian noise in the measurement input

33

HIL tests have smaller absolute mean and smaller standard deviation due to

Wire losses. Short lines = filter

Analysis of Real PMU data (telemetry):combined effect of measurement and process noise

34

0 500 1000 1500 2000 2500 3000 3500 40002.38

2.39

2.4x 10

5

time (s)

|V

| (

Volt

)

Voltage magnitude

0 500 1000 1500 2000 2500 3000 3500 4000142

143

144

145

time (s)

(

deg

ree)

Voltage angle

Under normal operation condition, loads fluctuate continuously and

randomly, which results in trends (or moving averages) along the noisy

PMU streams.

Analysis of Real PMU data (telemetry):combined effect of measurement and process noise

In order to properly calculate the noise variance, this trend has to be

eliminated from the raw PMU data.

Proposed method:

There are many different curve fitting tools. Fourier 4 model is applied

here, where 4 illustrates the number of terms.

General model Fourier4:

ft(a0,a1,b1,a2,b2,...,a4,b4,w,x) = a0 + a1*cos(x*w) + b1*sin(x*w) + a2*cos(2*x*w) +

b2*sin(2*x*w) + a3*cos(3*x*w) + b3*sin(3*x*w) + a4*cos(4*x*w) + b4*sin(4*x*w)

35

Under normal operation condition, loads fluctuate continuously and

randomly, which results in trends (or moving averages) along the noisy

PMU streams.

Raw dataCurve

fitting

Detrended

data

pdf pdf pdf

Results for the real PMU data test

36

Combined effect of both

process and measurement noise

0 1000 2000 3000 40002.38

2.39

2.4x 10

5

raw |V|

trend

2.38 2.385 2.39 2.395 2.4

x 105

0

5000

10000

histogram raw |V|

0 1000 2000 3000 40002.386

2.388

2.39

2.392x 10

5

trend

2.386 2.388 2.39 2.392

x 105

0

5000

10000

15000

histogram trend

0 1000 2000 3000 4000-1000

0

1000

detrend |V|

-1000 -500 0 500 10000

5000

10000

histogram detrend |V|

Results for the real PMU data test

37

-800 -600 -400 -200 0 200 400 600 800 10000

2000

4000

6000

8000

10000

12000distribution fit for the detrended data

histogram detrend data

normalDist fit

cauchyDist fit

laplaceDist fit

GoodnessOfFit function in Matlab

returns the goodness of fit between the

data and the reference.

Cost function: MSE( Mean square error)

fit_normal = 4.4142e+04

fit_cauchy = 6.1520e+08

fit_laplace = 2.2067e+04

2ref

s

x xfit

N

Conclusions & Further work I

Only under measurement noise, will the pdf be Gaussian .

Not only under measurement noise, but also with e.g. harmonics, unbalanced

three-phase, the pdf will have a different/biased expected value .

However the weights will not reflect that since the does not change.

Larger measurement error than expected

Further work: How should be the measurement equations weighted by taking

into account this knowledge.

38

( , )

'

Variance is not enough to take into account measurement errors due to

measurement noise under different impairments.

Conclusions & Further work II

The process noise (i.e. random load variations and resulting system

response), influences the different types of pdfs.

We cannot conclude that process noise alone is the contributing

factor because we are observing the combined/coupled effect of

both process and measurement noise.

Further work: carry out the off-line and RT-HIL simulations under

stochastic variations.

39

Real PMU Data: histograms do not look like Gaussian distributions!

Engineering

Phasor State Estimation Weighting Coefficients for AC and Hybrid Networks with Power Electronic Devices - or - How to Quantify Measurements Weights from PMUs?