October 28, 2008 Final Work Presentation 1

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Nov 2, 2013 (3 years and 9 months ago)

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October 28, 2008

Final Work Presentation

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October 28, 2008

Final Work Presentation

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Introduction

Application
-

ESP


Electro Static Precipitator or Electrostatic air cleaner


-

ESP is used to filter and remove particles from a flowing gas


-

ESP contains rows of thin wires(electrodes) situated between large metal
plates(collection electrodes)


-

High DC voltage between wires and plates


-

The particles are ionized by high voltage around the wires(cathode) and
then attracted by the plates(anode)


-

The load in the ESP application shows capacitive behaviour



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Introduction

DC/DC Power Converters


The need to convert a direct current(DC) voltage source from
one voltage level to another


Linear conversion in compare to Switched Mode conversion


A DC/DC switching power converter stores the input energy
temporarily and then releases that energy to the output at a
different voltage level (this function motivates the switching
property of the converter)


The storage may be in magnetic components, like inductors
and transformers, or in capacitors


Different topologies has been introduced to fulfill our
expectations, like Buck, Boost, Buck
-
Boost, Resonant power
converter and etc.




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Introduction

ESP application and DC/DC power converters


The proper high load voltage is
provided by a combination of a
DC/DC power converter and a
transformer


The resonant converter topology is
chosen for the power converter


The switching is done by the four
transistors


(Transistors can be used as switches
if they are saturated or cut
-
off)


Resonance circuit , switched system
and rectifier


Switching frequency and resonance
frequency


The load is connected in series to the
resonance circuit





Rectifier

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Introduction

Controller and Observer


Controller and Observer in a resonant converter


Resonant phase plane controller



Control Signal:


States:









Switched System(rectifier)

Resonant circuit

W
-
1

-

neg. conducting

W
0

-

not conducting

W
1

-

pos. conducting

Resonant current

Resonant voltage

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Modeling of dynamic systems


Why modelling is needed? Analysis of system properties, controller design
and observer design is based on the
dynamic model

of the system



The traditional differential equation form,
ABCD form
, A set of equations
derived from physical law governing the system




The ABCD form is not global for
non
-
linear systems
and it eliminates the

algebraic equations
(some information of the system are lost)



Hamiltonian modelling
is based on the energy properties of the system



Hamiltonian modeling DE
-
form and DAE
-
form


Power converter is a non
-
linear system due to existence of the switching
transistors and rectifier, this motivates using Hamiltonian modelling





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Hamiltonian Modelling








is the Hamiltonian states


is the energy storage function (Hamiltonian function)



is a skew
-
symmetric matrix models how energy flows within the system,


is a positive semi
-
definite symmetric dissipation matrix



is a skew
-
symmetric control matrix


*This model structure is used for modelling the converter*




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Hamiltonian Modelling


Hamiltonian states, :
charge on the capacitors, , and the magnetic
flow in the inductors,


The energy function:






where and



The physical interpretation of









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Hamiltonian modelling of the converter in DE
-
form


Decreasing the number of the states to four. magnetising inductance , , is
high compared to the resonance inductance, making the magnetising
current comparably small and possible to neglect.


Hamiltonian modelling in DE
-
form





Hamiltonian modelling is strongly related to the
Graph Theory


Different block matrices in Hamiltonian DE
-
form must be defined
according to graph theory


A Hamiltonian observer can be defined based on 4th order Hamiltonian
model of the precess






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4th order Hamiltonian model of the converter


Power converter is a switching system due
to existence of the transistors and rectifier


The controller produce the proper function
to switch the transistors


The rectifier has 3 possible conducting
states: Positive conducting, Non
-
conducting and Negative conducting



Rectifier is acting in which subspase!?


Ω1, positive conduction:



Ω0, no conduction:




-
1, negative conduction:







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Modelling for subspace

0



The graph
in the subspase

0


Spanning tree, tree branches and links


Component matrices, intermediate block matrices


and the system block matrices are defined according


to the graph theory








The intermediate block matrices

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Modelling for subspace

0


The system block matrices










The system equation:



= +


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Modelling for the subspaces

1 and

-
1


In the subspaces

1 and

-
1 the converter can be modelled by the graphs







Different matrices must be defined for the upper configurations, the final result:



where:









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4th order global model



A global model is:




The 4th order global model of the converter system




where




An Hamiltonian observer can be introduced based on the 4th order global model



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Hamiltonian Observer


The structure of an Hamiltonian observer is given by









From the process model and the observer structure, the estimation error
is given as , satisfies



The error can be regarded as a

. Hamiltonian system!!




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Hamiltonian Observer


Error in the observer can be regarded as a Hamiltonian process with the
Hamiltonian function(energy function):




it is shown that (Hultgren, lenells 2004) : if



Note: The process is modelled by a 4th order Hamiltonian model and the
observer is based on the 4th order Hamiltonian model of the process

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Discretization of the observer


Controller creates control signal from the external signal (reference)
and (estimation of the states)







Implementation of the controller: Analog or Digital


In the converter a phase plane feedback controller is used, this controller
uses Resonance voltage and resonance current as feedbacks



The only measurment in this application is the resonance current


In discretization of the Hamiltonian observer, the sampling interval is very
important (the error balance must be always negative)


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Implementation in Matlab and Simulink

Observer

Process

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Implementation in Matlab and Simulink

System Model

Hamiltonian Observer

Observer Model

Logic

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Working Plane


Normalization of voltages and currents!?


Where is the operating point practically?


Does the observer works


the same in all regions?


Where maximum error in


state estimation happens?


Does the observer get


unstable anywhere?


Which state has the most error?


How we can improve state


estimation in the observer?


Different modelling? Optimal Sampling frequency?




Result of simulation will answer the questions, hopefully correct!

iLN

vLN

High load current and

low load voltage

Low load current and
high load voltage

High Power

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High load current and Low load voltage, 10MHz sampling frequency

Spikes! High error!

High load current

Square Wave


Different subspaces!

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High load current and Low load voltage, 30MHz sampling frequency

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High load current and Low load voltage, 3MHz sampling frequency

Error does not

converge to zero!

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Hamiltonian observer gain



How we can improve state estimation? Applying obser gain!? Changing the model!?


Applying an observer gain may improve state estimation but not always!










Lets have a look at trajectories:




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Low load current and high load voltage, 30MHz sampling frequency

Rectifier is operating mostly

In the non
-
conducting mode

Spikes are much lower

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High load current and high load voltage (operating point), 10MHz sampling
frequency


K=0







K
>0

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High load current and high load voltage (operating point),

30MHz sampling frequency


K=0








K
>0

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3
rd

order model of the process


If we have a lower order model of the system, implementation will be
simpler and we have less calculation in the observer


By less calculation in the observer, we can discretize the observer with
higher sampling frequency


Which state is candidate to be eliminated
?


The load capacitance voltage is only slowly varying due to the high value
of . During one conduction period of the rectifier the load voltage can
be viewed as a constant voltage source, , in series with the voltage


The influence of the parallel capacitance, , is only significant when the
rectifier is not conducting. In the case when the rectifier is conducting the
parallel capacitance is connected in parallel with much larger load
capacitance .During the periods when the rectifier is not conducting,
the voltage of the parallel capacitance normally is commuting from to
-

or vice versa. The length of the non
-
conducting time periods is only
significant when the load current is small.


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3
rd

order model of the process



subspace

0







subspaces

1 and

-
1


System equation

Global system equation

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High load current and Low load voltage, 10MHz sampling frequency

K=0

K
>0

In this region, the rectifier is in the non
-
conducting mode for a short time


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Low load current and High load voltage, 10MHz sampling frequency

In this region, the rectifier is in the non
-
conducting mode for most of the time


K=0

K
>0

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High load current and high load voltage (operating point),

30MHz sampling frequency

K=0

K
>0

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Modelling of the system by considering resistance across the load capacitance

The load in this industrial application shows

obvious capacitive behaviour, but due to
existence of corona current around the wires,
the load shows resistivity behaviour too

The modelling takes the same steps as we did for

previous cases. First we define different subspaces

, then we achieve the system equation according to


Hamiltonian modelling and graph theory

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Modelling of the system by considering resistance across the load capacitance

subspace Ω0

=

+

subspaces Ω1 and Ω
-
1

where

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Thank you for your attention!