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

1. 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)
wei3@kth.se luigi.vanfretti@gmail.com
1
IEEE PES General Meeting July 20, 2017, Chicago, USA

2. 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

3. 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

4. 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 x
e
ε
n m
e ¡ ie i iw i
e
( ) n
h x ¡
m
ε ¡
,l
l m x ¡
| |,| |, ,V I θ δ
{ { {
ˆ
i i iV m
measurementerror state
V V  

5. 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 x
e
ε

6. 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
s
k k k k
T
   kh x x x g x g x
Numerically solve differential
equations, i.e., numerical integration.
$ $
..
11 ( )
2
s
k k kk
T
  x x x x
Generalized form

7. Model example: STATCOM
7
1
K
T s 
ref
V
V

 stI
1
| | ( | |) | |ref
st st
K
I V V I
T T
  &
$  1 1( ): ( ) ( ) .
2
s
k k k k
T
   kh x x x g x g x
using
Pseudo-dynamic model :
refV
V
sX
stIstI max
cap
I max
ind
I
Capacitive Inductive
( ) :| | | | ref
s stV X I V h x
,
, 1 1
ˆ ˆ( ) :(1 ) | | | |
2 2
(1 ) | | | |.
2 2
s s
st k k
refs s s
st k k
T T K
I V
T T
T K T T K
V I V
T T T
 
  
  
kh x
 | |,| |, , ,| |
T
 stx 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:

8. 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 8
0
0.2
0.4
time
|I|(p.u.)
Imag-true
Imag-m
Imag-est
1 2 3 4 5 6 7 8
0
1
2
3
x 10
-16
time
Error(p.u.)
Imag-residual-error
1 2 3 4 5 6 7 8
0
0.2
0.4
time
|I|(p.u.)
Imag-true
Imag-m
Imag-est
1 2 3 4 5 6 7 8
0
1
2
3
x 10
-16
time
Error(p.u.)
Imag-residual-error

9. 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 8
0.1
0.15
0.2
0.25
time|Ist|(p.u.)
|Ist|-true
|Ist|-est
1 2 3 4 5 6 7 8
0
0.5
1
x 10
-13
time
Error(p.u.)
|Ist|-residual-error
1 2 3 4 5 6 7 8
0
0.2
0.4
time
|Ist|(p.u.)
|Ist|-ture
|Ist|-est
1 2 3 4 5 6 7 8
0
0.1
0.2
time
Error(p.u.)
|Ist|-residual-error
1.95 2 2.05 2.1
0.1
0.2
0.3
time
|Ist|(p.u.)
|Ist|-m
|Ist|-est0 2 4 6 8
0
0.05
0.1
0.15
0.2
time
Error(p.u.)
|Ist|-residual-error

10. 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

11. 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
1
i
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.

12. 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
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
33

/ 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 50 100
0
100
200
300
400
500
histogram PMU |V|
histogram refPMU |V|
pdf PMU |V|
pdf refPMU |V|
-100 -50 0 50 100
0
100
200
300
400
500
histogram PMU 
histogram refPMU 
pdf PMU 
pdf refPMU 

13. 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
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
33

/ 0
50*32 Hz
50Hz
Perfect 3 phase signals – histogram shows a peak at mean.
Assumption of perfect measurement  weights equal to 1.

14. 3-phase signals generation
With different Gaussian noise levels. For instance, 10% variation, 0 gain
0.8 0.9 1 1.1 1.2 1.3
0
2
4
6
8
10
12
histogram PMU |V|
histogram refPMU |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
33

/ 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 10
0
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
2
1 1.5 2
0.5
1
1.5
refPMU |V|
1 1.5 2
-5
0
5
10
refPMU 
1 1.5 2
0.8
1
1.2
PMU |V|
1 1.5 2
-5
0
5
PMU 

15. Summary of cases with measurement noise
15
0 0.1 0.2 0.3 0.4 0.5
1
1.002
1.004
1.006
1.008
 of the input noise
oftheoutput
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
oftheoutput
Angle
refPMU noise
PMU noise
0 0.1 0.2 0.3 0.4 0.5
0
0.05
0.1
0.15
0.2
 of the input noise
oftheoutput
Magnitude
refPMU noise
PMU noise
0 0.1 0.2 0.3 0.4 0.5
0
2
4
6
 of the input noise
oftheoutput
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.

16. Summary of cases with measurement noise
16
0 0.1 0.2 0.3 0.4 0.5
1
1.002
1.004
1.006
1.008
 of the input noise
oftheoutput
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
oftheoutput
Angle
refPMU noise
PMU noise
0 0.1 0.2 0.3 0.4 0.5
0
0.05
0.1
0.15
0.2
 of the input noise
oftheoutput
Magnitude
refPMU noise
PMU noise
0 0.1 0.2 0.3 0.4 0.5
0
2
4
6
 of the input noise
oftheoutput
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.

17. 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
33

/ 0
50*32 Hz
50Hz
3rd
harmonics

Introducing harmonics
17
3-phase signals generation
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 1.06
0
100
200
300
400
histogram PMU |V|
histogram refPMU |V|
pdf PMU |V|
pdf refPMU |V|
-10 -5 0 5 10
0
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 

18. 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
33

/ 0
50*32 Hz
50Hz
3rd
harmonics

Introducing harmonics
18
Under perfect condition, harmonics are filtered by the PMUs, which is
expected from design.

19. 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
33

/ 0
50*32 Hz
50Hz
Gaussian
noise

3rd
harmonics

Harmonics + measurement noise
19
3-phase signals generation
3rd harmonics on three phases with 0.5 gain + Gaussian noise with 10% standard deviation
1 1.5 2
0.8
1
1.2
refPMU |V|
1 1.5 2
-5
0
5
10
refPMU 
1 1.5 2
0.8
1
1.2
PMU |V|
1 1.5 2
-5
0
5
PMU 
Ref PMU PMU
Simulation magnitude (1.00195, 0.0465553) (1.00202, 0.0468067)
Simulation angle (-0.16289, 2.56022) (-0.155349, 2.0292)
0.8 0.9 1 1.1 1.2 1.3
0
5
10
15
histogram PMU |V|
histogram refPMU |V|
pdf PMU |V|
pdf refPMU |V|
-10 -5 0 5 10
0
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

20. Summary of harmonics + measurement noise
20
0 0.1 0.2 0.3 0.4 0.5
0
0.02
0.04
0.06
0.08
0.1
0.12
 of the input noise
oftheoutput
Magnitude
refPMU noise
refPMU noise and harmonics
PMU noise
PMU noise and harmonics
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
6
 of the input noise
oftheoutput
Angle
0 0.1 0.2 0.3 0.4 0.5
1
1.002
1.004
1.006
1.008
1.01
 of the input noise
oftheoutput
Magnitude
refPMU noise
refPMU noise and harmonics
PMU noise
PMU noise and harmonics
0 0.1 0.2 0.3 0.4 0.5
-0.4
-0.3
-0.2
-0.1
0
 of the input noise
oftheoutput
Angle
Introducing harmonics increases the absolute
mean values for both magnitude and angle.

21. Summary of harmonics + measurement noise
21
0 0.1 0.2 0.3 0.4 0.5
0
0.02
0.04
0.06
0.08
0.1
0.12
 of the input noise
oftheoutput
Magnitude
refPMU noise
refPMU noise and harmonics
PMU noise
PMU noise and harmonics
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
6
 of the input noise
oftheoutput
Angle
0 0.1 0.2 0.3 0.4 0.5
1
1.002
1.004
1.006
1.008
1.01
 of the input noise
oftheoutput
Magnitude
refPMU noise
refPMU noise and harmonics
PMU noise
PMU noise and harmonics
0 0.1 0.2 0.3 0.4 0.5
-0.4
-0.3
-0.2
-0.1
0
 of the input noise
oftheoutput
Angle
Introducing harmonics does not have much
effect on the variances for both magnitude
and angle.

22. signal
generation
1,0.95,1 pu
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
33

/ 0
50*32 Hz
50Hz
Introducing unbalanced 3Φ
22
0 2 4 6 8 10
0.9833
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 
3-phase signals generation
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 50 100
0
100
200
300
400
500
histogram PMU |V|
histogram refPMU |V|
pdf PMU |V|
pdf refPMU |V|
-100 -50 0 50 100
0
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

23. signal
generation
1,0.95,1 pu
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
33

/ 0
50*32 Hz
50Hz
Introducing unbalanced 3Φ
23
Under perfect condition, unbalanced three-phase only affects the magnitude of
PMU output

24. Unbalanced 3Φ + measurement noise
24
signal
generation
1,0.95,1 pu
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
33

/ 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 2
0.5
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 2
0.5
1
1.5
refPMU |V|
1 1.2 1.4 1.6 1.8 2
-10
0
10
refPMU 
3-phase signals generation
Unbalanced three phases with magnitude 1, 0.95, 1 for a, b, c phase, respectively +
Gaussian noise with 10% standard deviation
Ref PMU PMU
Simulation magnitude (0.9853, 0.0465542) (0.985368, 0.0468055)
Simulation angle (-0.165661, 2.60371) (-0.157974, 2.06365)
0.8 0.9 1 1.1 1.2 1.3
0
2
4
6
8
10
12
histogram PMU |V|
histogram refPMU |V|
pdf PMU |V|
pdf refPMU |V|
-10 -5 0 5 10
0
2
4
6
8
10
12
14
histogram PMU 
histogram refPMU 
pdf PMU 
pdf refPMU 

25. Summary of unbalanced 3Φ+ meas. noise
25
0 0.1 0.2 0.3 0.4 0.5
0
0.02
0.04
0.06
0.08
0.1
0.12
 of the input noise
oftheoutput
Magnitude
refPMU noise
refPMU noise and unbalanced 3
PMU noise
PMU noise and unbalanced 3
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
6
 of the input noise
oftheoutput
Angle
0 0.1 0.2 0.3 0.4 0.5
0.98
0.99
1
1.01
1.02
 of the input noise
oftheoutput
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
oftheoutput
Angle
Introducing unbalanced three-phase affects
the absolute mean values for both magnitude
and angle.

26. Summary of unbalanced 3Φ+ meas. noise
26
0 0.1 0.2 0.3 0.4 0.5
0
0.02
0.04
0.06
0.08
0.1
0.12
 of the input noise
oftheoutput
Magnitude
refPMU noise
refPMU noise and unbalanced 3
PMU noise
PMU noise and unbalanced 3
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
6
 of the input noise
oftheoutput
Angle
0 0.1 0.2 0.3 0.4 0.5
0.98
0.99
1
1.01
1.02
 of the input noise
oftheoutput
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
oftheoutput
Angle
Introducing unbalanced three-phase does
not have much effect on the variances for
both magnitude and angle.

27. 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
3
Read data
locally
PMU connection
tester

3


28. Hardware-in-the-loop test—set up
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
3
Read data
locally
PMU connection
tester

3


29. Hardware-in-the-loop test—set up
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
3
Read data
locally
PMU connection
tester

3


30. Hardware-in-the-loop test—set up
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
3
Read data
locally
PMU connection
tester

3


31. Hardware-in-the-loop test—set up
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
3
Read data
locally
PMU connection
tester

3


32. Results of HIL test
Gaussian noise in the measurement input
32
0 0.1 0.2 0.3 0.4 0.5
0.97
0.98
0.99
1
1.01
1.02
 of the input noise
oftheoutput
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
oftheoutput
Angle
offline noise
HIL noise
0 0.1 0.2 0.3 0.4 0.5
0
0.05
0.1
0.15
0.2
 of the input noise
oftheoutput
Magnitude
offline noise
HIL noise
0 0.1 0.2 0.3 0.4 0.5
0
2
4
6
 of the input noise
oftheoutput
Angle
offline noise
HIL noise

33. Results of HIL test
Gaussian noise in the measurement input
33
HIL tests have smaller absolute mean and smaller standard deviation due to
 Wire losses. Short lines = filter

34. Analysis of Real PMU data (telemetry):
combined effect of measurement and process noise
34
0 500 1000 1500 2000 2500 3000 3500 4000
2.38
2.39
2.4
x 10
5
time (s)
|V|(Volt)
Voltage magnitude
0 500 1000 1500 2000 2500 3000 3500 4000
142
143
144
145
time (s)
(degree)
Voltage angle
Under normal operation condition, loads fluctuate continuously and
randomly, which results in trends (or moving averages) along the noisy
PMU streams.

35. 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 data
Curve
fitting
Detrended
data

pdf pdf pdf

36. Results for the real PMU data test
36
Combined effect of both
process and measurement noise
0 1000 2000 3000 4000
2.38
2.39
2.4
x 10
5
raw |V|
trend
2.38 2.385 2.39 2.395 2.4
x 10
5
0
5000
10000 histogram raw |V|
0 1000 2000 3000 4000
2.386
2.388
2.39
2.392
x 10
5
trend
2.386 2.388 2.39 2.392
x 10
5
0
5000
10000
15000
histogram trend
0 1000 2000 3000 4000
-1000
0
1000
detrend |V|
-1000 -500 0 500 1000
0
5000
10000 histogram detrend |V|

37. Results for the real PMU data test
37
-800 -600 -400 -200 0 200 400 600 800 1000
0
2000
4000
6000
8000
10000
12000
distribution 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 x
fit
N



38. 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.

39. 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!