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In front of the jury: M. Moamar Sayed-Mouchaweh
M. Laurent Rambault
M. Malek Ghanes
M. Edouard Laroche
M. Gilles Hermann
M. Patrice Wira
by Anh Tuan Phan
Laboratoire MIPS, Université de Haute Alsace
1
PhD Defense 16 September 2016
Power Systems Model Developments for Power
Quality Monitoring: Application to Fundamental
Frequency and Unbalance Estimation
Power quality problems:-fundamental frequency deviation-unbalance
-harmonics-voltage swell/sag
photovoltaicpanel
windgenerator
eletrical
car
home
generation
... ...
power plant
engine
other loadsother
generator
consumptiontransmission and distribution
2
A.T. Phan PhD defense
Power quality is a central issue for the whole
grid’s reliability
3
The research of the thesis aims to improve the power quality of power systems
A.T. Phan PhD defense
1. Power quality disturbances and monitoring of power quality
2. Well-known methods for power quality improvement
3. New state-space representations for frequency estimation
4. New state-space representations for symmetrical
components identification
5. Conclusions and perspectives
Content
4
A.T. Phan PhD defense
1. Power quality disturbances and monitoring of power quality
I. Power quality problems
II. Power quality control
III. Problematic
2. Well-known methods for power quality improvement
3. New state-space representations for frequency estimation
4. New state-space representations for symmetrical
components identification
5. Conclusions and perspectives
Content
5
A.T. Phan PhD defense
Ideal three phase signals of a three phase
power system
6
Principle of an ideal three phase generator:
Current waveforms: ( ) sin( )
( ) sin( 2 / 3)
( ) sin( 2 / 3)
a
b
c
i t I t
i t I t
i t I t
1. Power quality disturbances and
monitoring of power quality
A.T. Phan PhD defense
Power quality problems:
Unbalance of three phase systems
7
Bala
nce
Unbala
nce
( ) sin( )
( ) sin( 2 / 3)
( ) sin( 2 / 3)
a
b
c
i t I t
i t I t
i t I t
( ) sin( )
( ) sin( )
( ) sin( )
a
b
c
a
b
a
b
c c
i t t
i t t
i It t
I
I
Consequences:
1. power losses, heating, reduced productivity and vibration to asynchronous
motors and synchronous generators.
2. limited line transmission capacity because of additional heating.
1. Power quality disturbances and
monitoring of power quality
A.T. Phan PhD defense
Power quality problems:
Frequency deviation
8
The value of the fundamental frequency depends on the equilibrium of the
power generation and the power demand.
Power generation Power demand
Ideally:
generated power = consumed power
The frequency rises if the power
generation is higher than the power
demand.
The frequency falls otherwise.
50Hzof
Sou
rce
: ww
w.v
ente
ea.f
r 50.87Hzof
49.19Hzof
1. Power quality disturbances and
monitoring of power quality
A.T. Phan PhD defense
The effects of frequency deviation
• Large errors occur in protective relays due to the frequency variation
• The magnetic characteristic of transformers can get into non-linear zones
when the fundamental frequency varies in time.
9
• The operation of rotating machinery, or processes using their timing from the
power frequency will be affected when the frequency changes
1. Power quality disturbances and
monitoring of power quality
Frequency variation has great impact to normal operation of electrical devices,
among them:
A.T. Phan PhD defense
0 100 200 300-8
-6
-4
-2
0
2
4
6
8Distorted current waveform
0 100 200 300-8
-6
-4
-2
0
2
4
6
8Equivalent harmonic components
fundamental
7th harmonic
11th harmonic
Power quality problems:
Harmonics
10
1. Power quality disturbances and
monitoring of power quality
A.T. Phan PhD defense
The effects of harmonics
• Heating in electrical machines
• Insulation failure in power cables
• Decrease in response speed and mis-operation of relays
• Mis-operation and/or malfunction in electronic equipment
11
The propagation of harmonics in power systems leads to many negative
effects such as:
1. Power quality disturbances and
monitoring of power quality
A.T. Phan PhD defense
Power quality control:
Active power filter
12
automatic and control power
electronics signal
processing
POWER LINE
SHUNT ACTIVE FIL TER
NONLINEAR
LOAD
CONTROL LAW
PARAMETER
ESTIMATION
POWER SOURCE
INVERTER
measured line current
compensating current
harmonic
components
reference
signal
1. Power quality disturbances and
monitoring of power quality
A.T. Phan PhD defense
Objective of the thesis
13
automatic and control power
electronics signal
processing
POWER LINE
SHUNT ACTIVE FIL TER
NONLINEAR
LOAD
CONTROL LAW
PARAMETER
ESTIMATION
POWER SOURCE
INVERTER
measured line current
compensating current
harmonic
components
reference
signal • Frequency estimation
• Symmetrical components
estimation
• Harmonic current estimation
• Reactive power estimation
signal processing
PARAMETER
ESTIMATION
1. Power quality disturbances and
monitoring of power quality
A.T. Phan PhD defense
1. Power quality disturbances and monitoring of power quality
2. Well-known methods for power quality improvement
I. Extended Kalman Filter for balanced three phase power
signals (3P EKF)
II. Extended Kalman Filter for one phase power signal (1P EKF)
3. New state-space representations for frequency estimation
4. New state-space representations for symmetrical
components identification
5. Conclusions and perspectives
Content
14
A.T. Phan PhD defense
Extended Kalman Filter for balanced three
phase power signals (3P EKF)
( ) sin( )
( ) sin( 2 / 3)
( ) sin( 2 / 3)
a s
b s
c s
i k I kT
i k I kT
i k I kT
1( ) sj kTy k A e
Clark’s transform
15
2. Well-known methods for power
quality improvement
• Clark’s transform: 2 1 1
3 2 2
1( )
2
( ) ( ) ( ) ( )a b c
b c
i i i i
i i i
k k k k
( ) ( ) ( )y k i jik k
• Complex representation:
power
system
A.T. Phan PhD defense
Extended Kalman Filter for balanced three
phase power signals (3P EKF)
( ) sin( )
( ) sin( 2 / 3)
( ) sin( 2 / 3)
a s
b s
c s
i k I kT
i k I kT
i k I kT
1( ) sj kTy k A e
Clark’s transform
1. Selection of the state variables
1
2 ( ) : sj kTq k Ae
1( ) : sj Tq k e
2. Definition of the state-space representation
1 1
2 1 2
( 1) 1 0 ( )
( 1) 0 ( ) ( )
q k q k
q k q k q k
2( ) ( )y k q k
16
2. Well-known methods for power
quality improvement
A.T. Phan PhD defense
Extended Kalman Filter for one phase power
signal (1P EKF)
( ) sin( ) 0.5 0.5 s sj kT j kTsi k I kT j Ie j Ie
17
2. Well-known methods for power
quality improvement
A.T. Phan PhD defense
Extended Kalman Filter for one phase power
signal (1P EKF)
( ) sin( ) 0.5 0.5 s sj kT j kTsi k I kT j Ie j Ie
18
1. Selection of the state variables
1( ) : sj Tq k e
2( ): sj kT jq k Ie
1 1
2 21
1
13 3
( 1) ( )1 0 0
( 1) ( )0 ( ) 0
0 0 ( )( 1) ( )
q k q k
q k q kq k
q kq k q k
2 3( ) ( ) 0.5 ( ) 0.5 ( )y k i k j q k j q k
3:( ) sj kT j
q Iek
2. Well-known methods for power
quality improvement
2. Definition of the state-space representation
A.T. Phan PhD defense
Characteristics of the well-known methods
Methods Characteristics
Underlying model Advantages Disadvantages Applications
3P EKF Nonlinear state-
space model
Sample based,
robust to noise
- Difficult to choose
initial values of
the state
variables.
- Balanced three
phase signals
1P EKF Nonlinear state-
space model
Sample based,
robust to noise
- Difficult to choose
initial values of
the state
variables.
- Single phase
signals
19
2. Well-known methods for power
quality improvement
Comments:
• The models of these two methods are not suitable to unbalanced systems
• We have compared them with Adaptive Prony’s method and Adaptive
Notch Filter in [1]
A.T. Phan PhD defense 2. Well-known methods for power
quality improvement
Discussion
20
Balanced
three
phase
systems
One phase
signals
Unbalanced three
phase systems
Main problems:
• Frequency fluctuation
• Harmonics
• Unbalance
• Reactive power
Tools and solutions:
• Symmetrical
Components
• Our proposed
state-space models
Main problems:
• Frequency fluctuation
• Harmonics
• Reactive power
Tools and solutions:
• 3P EKF
Main problems:
• Frequency fluctuation
• Harmonics
• Reactive power
Tools and solutions:
• 1P EKF
• Adaptive Notch Filter
• Adaptive Prony’s
method
One
phase
systems
A.T. Phan PhD defense
1. Power quality disturbances and monitoring of power quality
2. Well-known methods for power quality improvement
3. New state-space representations for frequency estimation
I. Theory of symmetrical components
II. New state-space representations and algorithms
III. Initialization scheme for the state variables
IV. Simulation tests and results
4. New state-space representations for symmetrical
components identification
5. Conclusions and perspectives
Content
21
A.T. Phan PhD defense
Analysis tools for unbalanced three phase
signals: symmetrical components
aI
bI
cI
120o
120o120o
aI
cI
bI
120o
120o120oo
bIo
aI
ocI
22 3. New state-space representations
for frequency estimation
positive components
(balanced three
phase system)
negative components
(balanced three
phase system)
zero components
(symmetrical)
aI
bI
cI
An unbalanced three
phase system
Any unbalanced three phase systems can be represented as a unique sum of
three symmetrical components:
A.T. Phan PhD defense
New state-space models modeling unbalanced
three phase systems
( )
( )
( )
( ) sin
( ) sin
( ) sin
a s
b s
c s
b
a
c
a
c
b
I
I
kT
kT
kTI
i k
i k
i k
( ) 0s sj kT j j kT jy k A e A e
23
Clark’s transform
power system
3. New state-space representations
for frequency estimation
A.T. Phan PhD defense
New state-space models modeling unbalanced
three phase systems
( ) 0s sj kT j j kT jy k A e A e
24
Clark’s transform ( )
( )
( )
( ) sin
( ) sin
( ) sin
a s
b s
c s
b
a
c
a
c
b
I
I
kT
kT
kTI
i k
i k
i k
3. New state-space representations
for frequency estimation
A.T. Phan PhD defense
New state-space models modeling unbalanced
three phase systems [4]
( ) 0s sj kT j j kT jy k A e A e
25
1. Selection of the state variables
1( ) : sj Tq k e
2. Definition of a state-space representation
1 1
2 21
1
13 3
( 1) ( )1 0 0
( 1) ( )0 ( ) 0
0 0 ( )( 1) ( )
q k q k
q k q kq k
q kq k q k
2 3( ) ( ) ( )y k q k q k
Clark’s transform
2( ) sj kT j
q A ek
3( ) sj kT j
q A ek
( )
( )
( )
( ) sin
( ) sin
( ) sin
a s
b s
c s
b
a
c
a
c
b
I
I
kT
kT
kTI
i k
i k
i k
3. New state-space representations
for frequency estimation
A.T. Phan PhD defense
Properties of the new state-space model
Meaning of the
state variables
Model of 3P
EKF
Model of 1P
EKF
Proposed
model
Represents the
fundamental
frequency
Represents the
fundamental
frequency
Represents the
fundamental
frequency
Represents the
positive
components
No physical
meaning
Represents the
positive
components
Does not exist No physical
meaning
Represents the
negative
components
1( )q k
2( )q k
3( )q k
26
3. New state-space representations
for frequency estimation
The proposed model and its properties have been presented in [3].
A.T. Phan PhD defense
Method to estimate the fundamental frequency
using the proposed model: Extended Kalman Filter
11
1ˆ ( ) sin img( ( ))2
o
s
f k q kT
27
Iterative
estimator
3. New state-space representations
for frequency estimation
This approach has been published in a conference [4]
A.T. Phan PhD defense
How to initialize the state variables in EKF?
28
Assign ( )
a value near
the nominal value
Estimate states ,
in several iterations
by Kalman Filter (KF)
Use the estimated
state variables
for tracking stage
3q
of
2q
1
2
3
1q
2 2
3 3
0( 1) ( )
1( 1) ( )0
aq k q k
q k q ka
2 3( ) ( ) ( )y k q k q k
1( ) sj Ta kq e
State-space model when is
assigned a fixed value: of
3. New state-space representations
for frequency estimation
A.T. Phan PhD defense
Algorithm diagram of the new proposed
method to estimate the fundamental frequency
12
0
1ˆ( ) [ ( ) ( )]
L
i
k y k i y k iL
1 1
2 21
3 3
1
( 1) ( )1 0 0
( 1) ( )0 ( ) 0
1( 1) ( )0 0
( )
q k q k
q k q kq k
q k q k
q k
2 3( ) ( ) ( )y k q k q k
Initialization stage
Tracking stage
( )k threshold
?
Y
N
State-space representation of unbalanced
three phase systems:
Definition of : ( )k
29 3. New state-space representations
for frequency estimation
A.T. Phan PhD defense
Simulation tests and results
a) Unbalanced three phase sinusoidal signals
b) Unbalanced three phase sinusoidal signals disturbed by 30 dB noise
c) Unbalanced three phase sinusoidal signals disturbed by harmonics
a) b) c)
30
I. Performance of the proposed method compared to 1P EKF and 3P EKF
(without employing the proposed initialization scheme) in estimating
the fundamental frequency of:
3. New state-space representations
for frequency estimation
A.T. Phan PhD defense
Results of estimating fundamental
frequency of an unbalanced system a)
3. New state-space representations
for frequency estimation
31
A.T. Phan PhD defense
Results of estimating fundamental
frequency of an unbalanced system
Methods time to reach the
reference
frequency with
+/- 0.1 Hz (ms)
MSE at steady-
state
3P EKF 100 10−4
1P EKF 57 10−6
Proposed method 49 10−7
a)
3. New state-space representations
for frequency estimation
32
* The time of one period of the signal is 20 (ms)
A.T. Phan PhD defense
Results of estimating fundamental frequency
of an unbalanced system with an additional
30 dB noise
33
b)
3. New state-space representations
for frequency estimation
A.T. Phan PhD defense
Results of estimating fundamental frequency
of an unbalanced system with an additional
30 dB noise
34
b)
3. New state-space representations
for frequency estimation
Methods time to reach the
reference
frequency with
+/- 0.1 Hz (ms)
MSE at steady-
state
3P EKF 115 10−4
1P EKF 47 10−6
Proposed method 49 10−6
* The time of one period of the signal is 20 (ms)
A.T. Phan PhD defense
Results of estimating fundamental frequency
of an unbalanced system disturbed by
harmonics
35
c)
3. New state-space representations
for frequency estimation
Phase A Phase B Phase C Phase
sequence
0° 120° 240° A-B-C
3x0° (0°)
3x120° (360°=0°)
3x240° (720°=0°)
In phase
5x0° (0°)
5x120° (-120°)
5x240° (-240°)
C-B-A
7x0°
(0°) 7x120° (120°)
7x240°
(240°)
A-B-C
Fundamental
3rd harmonic
5th harmonic
7th harmonic
sou
rce:
ww
w.a
llab
ou
tcir
cuit
s.co
m
This table shows the harmonic phase sequences of some harmonics
A.T. Phan PhD defense
Results of estimating fundamental frequency
of an unbalanced system disturbed by
harmonics
36
c)
3. New state-space representations
for frequency estimation
A.T. Phan PhD defense
Results of estimating fundamental frequency
of an unbalanced system disturbed by
harmonics
37
c)
3. New state-space representations
for frequency estimation
Methods time to reach the
reference
frequency with
+/- 0.1 Hz (ms)
MSE at steady-
state
3P EKF 135 10−4
1P EKF 150 10−4
Proposed method 70 10−6
* The time of one period of the signal is 20 (ms)
A.T. Phan PhD defense
Simulation tests and results
II. The performance of the proposed method combined with the initialization
scheme is evaluated in estimating the fundamental frequency for:
38
a) Unbalanced three phase sinusoidal signals with the frequency and
amplitudes experiencing step changes.
b) Unbalanced three phase sinusoidal signals with the frequency varying
as a sinusoidal wave.
3. New state-space representations
for frequency estimation
A.T. Phan PhD defense
Case a) Signals for test
50o
f 50.5o
f
1A 0.8A
0.2A 0.3A
Before load change After load change
time (s)
39
3. New state-space representations
for frequency estimation
A.T. Phan PhD defense
Results of estimating the fundamental
frequency
49.5(Hz)
45(Hz)
40
Initial value of the
fundamental frequency:
Initial value of the
fundamental frequency:
3. New state-space representations
for frequency estimation
A.T. Phan PhD defense
Results of estimating the fundamental
frequency
Initialization time to reach the
reference
frequency with
+/- 0.1 Hz (ms)
MSE at steady-
state
49.5 Hz 3 10−10
45 Hz 5.8 10−7
41
3. New state-space representations
for frequency estimation
* The time of one period of the signal is 20 (ms)
A.T. Phan PhD defense
Case b) Frequency tracking
The estimated frequency
and the real one
(the frequency is
initialized at 49.5 Hz)
The estimation error
42
3. New state-space representations
for frequency estimation
A.T. Phan PhD defense
Discussion
• The proposed method is an accurate frequency estimator even in time-
varying environments and unbalanced conditions.
• The initialization scheme helps to solve the problems of choosing initial
values for the state variables.
43
• The performance of 3P EKF is degraded when the system is unbalanced.
3. New state-space representations
for frequency estimation
• Unlike the other two methods, 1P EKF is unable to eliminate the impact of
the harmonics whose three phase signals are in phase with each others.
• The proposed nonlinear model has been published in [2].
A.T. Phan PhD defense
1. Power quality disturbances and monitoring of power quality
2. Well-known methods for power quality improvement
3. New state-space representations for frequency estimation
4. New state-space representations for symmetrical
components identification
I. New state-space representations and algorithms
II. Initialization scheme for the state variables
III. Simulation tests and results
5. Conclusions and perspectives
Content
44
A.T. Phan PhD defense
Clark’s transform of unbalanced three phase
signals
45
( ) 0s sj kT j j kT jy k A e A e
Clark’s transform
4. New state-space representations for
symmetrical components identification
( )
( )
( )
( ) sin
( ) sin
( ) sin
a s
b s
c s
b
a
c
a
c
b
I
I
kT
kT
kTI
i k
i k
i k
Power system
A.T. Phan PhD defense
New state-space models modeling unbalanced
three phase systems
Fundamental frequency
is unknown Fundamental frequency
is known
46
1 1
2 21
3 3
1
( 1) ( )1 0 0
( 1) ( )0 ( ) 0
1( 1) ( )0 0
( )
q k q k
q k q kq k
q k q k
q k
2 3( ) ( ) ( )y k q k q k
1( ) : sj Tkq e
Nonlinear model
1 2
Linear model
2 2
3 3
0( 1) ( )
1( 1) ( )0
aq k q k
q k q ka
2 3( ) ( ) ( )y k q k q k
1( ): sj Ta kq e
4. New state-space representations for
symmetrical components identification
The nonlinear model is published in [2]
A.T. Phan PhD defense
Methods to estimate the symmetrical components
using the proposed models: EKF and KF
2. The proposed linear state space model is combined with KF
abc 3 q
2 q
1 q abc
i
i Clark’s
transformation
inverse Clark’s transformation
symmetrical components
average
proposed nonlinear state space representation
EKF
, , a b c i i i
, , a b c i i i
, , a b c i i i
o o o
+ + +
- - -
state variables
measured line currents
, , a b c i i i
+
47
1. The proposed nonlinear state space model is combined with EKF
1
2
abc 3 q
2 q abc
i
i Clark’s
transformation
inverse Clark’s transformation
symmetrical components
average
proposed linear state space
representation KF
, , a b c i i i
, , a b c i i i
, , a b c i i i
o o o
+ + +
- - -
state variables
measured line currents
, , a b c i i i
+
of
4. New state-space representations for
symmetrical components identification
computing of
A.T. Phan PhD defense
Algorithm diagram of the new proposed method to
estimate the symmetrical components
Initialization stage
Tracking stage
( )k threshold
?
Y
N
48
4. New state-space representations for
symmetrical components identification
A.T. Phan PhD defense
Simulation tests and results
I. The fundamental frequency is known, the proposed method for this
case is used to estimate the symmetrical components of unbalanced three
phase sinusoidal signals. The results are compared with that of method
MO-Adaline.
49
2 2
3 3
0( 1) ( )
1( 1) ( )0
aq k q k
q k q ka
2 3( ) ( ) ( )y k q k q k
This is a linear model 1
4. New state-space representations for
symmetrical components identification
A.T. Phan PhD defense
Results of estimating the symmetrical components
of unbalanced three phase sinusoidal signals
The unbalanced three-
phase signals
Estimated amplitudes
of positive and
negative components
Estimated phase
angles of positive and
negative components
50
4. New state-space representations for
symmetrical components identification
A.T. Phan PhD defense
Results of estimating the symmetrical components
of unbalanced three phase sinusoidal signals
51
Amplitudes MSE at steady-
state
MO-Adaline 10−7
Proposed method 10−14
Amplitudes MSE at steady-
state
MO-Adaline 10−8
Proposed method 10−15
Positive
components:
Negative
components:
4. New state-space representations for
symmetrical components identification
A.T. Phan PhD defense
Simulation tests and results
a) Unbalanced sinusoidal three phase signals with the frequency and the
amplitudes experiencing step changes.
b) Unbalanced sinusoidal three phase signals during frequency variation.
52
II. The fundamental frequency is unknown, the proposed method for this case is used to estimate the symmetrical components of:
is a nonlinear model
1 1
2 21
3 3
1
( 1) ( )1 0 0
( 1) ( )0 ( ) 0
1( 1) ( )0 0
( )
q k q k
q k q kq k
q k q k
q k
2 3( ) ( ) ( )y k q k q k
2 4. New state-space representations for
symmetrical components identification
A.T. Phan PhD defense
Case a) Signals for test
50o
f 50.5o
f
1A 0.8A
0.2A 0.3A
Before load change After load change
time (s)
53
4. New state-space representations for
symmetrical components identification
A.T. Phan PhD defense
Estimation of the symmetrical components of
unbalanced three phase sinusoidal signals
experiencing step changes
Estimation error
of the positive
components
Estimation error
of the negative
components
54
4. New state-space representations for
symmetrical components identification
A.T. Phan PhD defense
4. New state-space representations for
symmetrical components identification
Case b) Estimation of the symmetrical components
of unbalanced three phase sinusoidal signals with
the frequency varying
55
Estimation error of
the positive
components
Estimation error of
the negative
components
A.T. Phan PhD defense
Discussion
• With the fundamental frequency supposed to be available, the linear
method is able to estimate effectively the three symmetrical
components.
• The nonlinear method , when combined with the proposed initialization scheme, is efficient in estimating the symmetrical components of time-varying unbalanced power systems.
56
1
2
4. New state-space representations for
symmetrical components identification
• The method and its applications in frequency and unbalance estimation
have been presented in a journal article to appear [5].
A.T. Phan PhD defense
1. Power quality disturbances and monitoring of power quality
2. Well-known methods for power quality improvement
3. New state-space representations for frequency estimation
4. New state-space representations for symmetrical
components identification
5. Conclusions and perspectives
Content
57
A.T. Phan PhD defense
Conclusions
• Power quality is a central issue for the whole grid’s reliability.
• Signal processing methods are useful to monitor and control of power
quality effectively, however, most of them concern one phase signals
and/or balanced three phase signals.
58 5. Conclusions and perspectives
A.T. Phan PhD defense
Conclusions
• The thesis proposes new methods to estimate the parameters of unbalanced
three phase power systems. The methods are based on new state-space
modeling of the unbalanced systems and applied to:
– Estimate the fundamental frequency of the power systems
– Estimate the symmetrical components of the power systems
59
• Simulation results have proven that the new methods are efficient and robust in estimating the fundamental frequency and the symmetrical components of a time-varying power system under various severe unbalanced conditions.
• The work developed during this thesis has been published in 4 international conferences and 1 international journal.
5. Conclusions and perspectives
A.T. Phan PhD defense
Perspectives
• Expand the proposed state-space models in order to take into account the
harmonic components, as each component of three phases could be
considered as a positive or negative sequence.
60 5. Conclusions and perspectives
• Look to associate to the proposed state space model another identification
algorithm, e.g., Artificial Neural Networks, to improve the performance of
the method in disturbed and time-varying environment of power systems.
• Enhance the proposed state-space models for higher order unbalanced
harmonics, the purpose is to come up to a general and uniform state-space
model able to include in one concept or model the unbalanced fundamental
component, the unbalanced harmonics, and the harmonic sequences.
A.T. Phan PhD defense
Publications and Authors’ contributions
61
Conference papers published in international conferences with review and proceedings:
[1]. Anh Tuan Phan, Gilles Hermann, and Patrice Wira. “Online Frequency Estimation in
Power Systems: A Comparative Study of Adaptive Methods”. In: 40th Annual Conference of the IEEE
Industrial Electronics Society (IECON 2014), Dallas, TX - USA, pages 4352–4357, 2014
[ 2]. Anh Tuan Phan, Gilles Hermann, and Patrice Wira. Kalman filtering with a new state-space model for
three-phase systems: Application to the identification of symmetrical components. In IEEE Conference on
Evolving and Adaptive Intelligent Systems (EAIS 2015), Douai-France, pages 216–221, 2015
[ 3]. Anh Tuan Phan, Duc Du Ho, Gilles Hermann, and Patrice Wira. A new state-space model for three-
phase systems for kalman filtering with application to power quality estimation. In 11th International
Conference of Computational Methods in Sciences and Engineering (ICCMSE 2015), Athens-Greece, 2015
[ 4]. Anh Tuan Phan, Gilles Hermann, and Patrice Wira. A new state-space for unbalanced
three-phase systems: Application to fundamental frequency tracking with kalman filtering. In 18th IEEE
Mediterranean Electrotechnical Conference (MELECON 2016), Limassol Cyprus, 2016
Article published in international journals:
[ 5]. Anh Tuan Phan, Patrice Wira, and Gilles Hermann. A Dedicated State Space for Power
System Modeling and Frequency and Unbalance Estimation, Evolving Systems, to appear in 2016
A.T. Phan PhD defense
Results of estimating the positive components
of an unbalanced system: load change
50o
f 50o
f
1.2A 1A
0A 0.2A
Before load change After load change
time (s)
63
A.T. Phan PhD defense
Results of estimating the positive components
of an unbalanced system: load change
Phase A
Phase B
Phase C
64
A.T. Phan PhD defense
Results of estimating the negative components
of an unbalanced system: load change
Phase A
Phase B
Phase C
65
A.T. Phan PhD defense
Adaptive Prony’s method
( )u k k
ˆ( )i k1
0 1
1
1
...( )
1 ...
m
m
n
n
b b z b zH z
a z a z
( )i k
( )e k
( )u k
update iaib
( )i k measured line current :
Input: Reference signal:
by Least Square Method
66
A.T. Phan PhD defense
Adaptive Notch Filter
1 2
1 2 2
1 2cos( )( , )
1 2cos( )
z zH z
z z
(1 )
Frequency response of
( )jH e
( , )H z
( )i k ( , )e k ( , )H z
measured line
current update
by Least Mean Square Method 67
A.T. Phan PhD defense
Algorithm
abc 3 q
2 q
1 q abc
i
i Clark’s
transformation
inverse Clark’s transformation
symmetrical components
average
proposed state space
representation KF
, , a b c i i i
, , a b c i i i
, , a b c i i i
o o o
+ + +
- - -
state variables
measured line currents
, , a b c i i i
+
69
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Tracking of a fundamental frequency varying
constantly in time
Real and estimated
frequency (Hz)
Frequency estimation
error (Hz)
70
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Power quality problems:
Frequency deviation
Fundamental frequency varies in time
71
A.T. Phan PhD defense
Power quality problems:
Frequency deviation
Sweden
Singapore
73 https://en.wikipedia.org
Central Europe
China
Great Britain
A.T. Phan PhD defense
Power quality problems:
Unbalance of three phase systems
74
Balance Unbalance
( ) sin( )
( ) sin( 2 / 3)
( ) sin( 2 / 3)
a
b
c
i t I t
i t I t
i t I t
( ) sin( )
( ) sin( )
( ) sin( )
a
b
c
a
b
a
b
c c
i t t
i t t
i It t
I
I
• power losses, heating, reduced productivity and vibration to
asynchronous motors and synchronous generators. • limited line transmission capacity because of additional heating.
Unbalance leads to:
A.T. Phan PhD defense
Algorithm to estimate state variables of the
proposed model: Extended Kalman Filter
11
1ˆ ( ) sin img( ( ))2
o
s
f k q kT
+
76
Iterative
estimator
A.T. Phan PhD defense
Results of estimating the symmetrical components
of unbalanced three phase sinusoidal signals
77
Amplitudes Mean at steady-
state
MSE at steady-
state
Error max.
at steady-
state
MO-Adaline 1.0000 10−7 10−5
Proposed method 0.9999 10−14 10−7
Amplitudes Mean at steady-
state
MSE at steady-
state
Error max.
at steady-
state
MO-Adaline 0.2000 10−8 0.0013
Proposed method 0.2000 10−15 10−7
Positive components:
Negative components:
A.T. Phan PhD defense
New state-space models modeling unbalanced
three phase systems
Measured
line currents
a b ci i i
Symmetrical component decomposition
State-space modeling
State-space modeling
State-space modeling
Model synthesis
a b ci i i a b ci i i
o o o
a b ci i i
78
zzzz 1 1
2 1 2
( 1) 1 0 ( )
( 1) 0 ( ) ( )
q k q k
q k q k q k
2( ) ( )y k q k
State space model of balanced
three phase signals:
3. New state-space representations
for frequency estimation
A.T. Phan PhD defense
Properties of the new state-space model
Characteristics Model of 3P EKF Model of 1P
EKF
Proposed
model
Linearity Nonlinear Nonlinear Nonlinear
Number of states 2 3 3
Output
Clark’s transform of
balanced three
phase signals
One phase
signal
Clark’s
transform of
unbalanced
three phase
signals
79
3. New state-space representations
for frequency estimation