Output high order sliding mode control of unity-power-factor in three-phase

AC/DC Boost Converter

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JianXing Liu, Salah Laghrouche, Maxim WackLaboratoire Systèmes Et Transports (SET)

Contents of the presentation

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Introduction

Problem Formulation

Second order sliding mode controller design

Second order sliding mode observer design

Simulation results with proposed controller

Conclusions

AC-DC power conversion

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AC-DC power conversion is required by all electronic devices

virtually

MOSFET, IGBT, are commonly used for AC-DC Converters

The electric utility grid has a sinusoidal waveform, and most electronic equipment needs a DC power supply.

AC-DC power conversion

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Power factor reflects the efficiency and quality of such process. Different control algorithms have been used to achieve unity power factor.

An output second order sliding mode control is designed here.

Problem:1. The performance and efficiency of power converters

with unknown varying load and internal uncertainties2. Minimize the number of the sensors

Model of three phase AC/DC converter

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Fig1 Three phase boost type AC/DC converter

The circuit of three phase AC/DC boost converterlinki

Model

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Model in Phase Coordinate Frame:101

1 1 2 3

2022 2 1 2

33 03 3 1 2

0 01 1 2 2 3 3

(2 )6

(2 )6

(2 )6

1 ( )2

g

g

g

l

UUdi r i u u udt L L L

UUdi r i u u udt L L L

Udi Ur i u u udt L L LdU U u i u i u idt R C C

= − − − − +

= − − − − + = − − − − + = − + + +

Model in (d,q) Coordinate Frame

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It is convenient to design the control in the rotating reference frame synchronized with the supply frequency.Transformation Matrix:

1

2

3

2 2cos( ) cos( ) cos( )2 3 32 23 sin( ) sin( ) sin( )3 3

d

q

xt t txx

xt t t x

ω ω π ω π

ω ω π ω π

− + = − − − − +

Define:2 2cos( ) cos( ) cos( )2 3 32 23 sin( ) sin( ) sin( )3 3

t t tC

t t t

ω ω π ω π

ω ω π ω π

− + =

− − − − +

Model in (d,q) Coordinate Frame

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Where ω is the source frequency

0

0

0 0

2

23 ( )

4

gddd q d

q gqq d q

d d q ql

Udi Ur i i udt L L Ldi UUr i i udt L L LdU U i u i udt R C C

ω

ω

= − + − +

= − − − + = − + +

Control objectives

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Unity power factor Ripple free output voltage with desired level

tracks , tracks Satisfy Power Balance Equation

di*di qi *

qi

* * 00

3 ( )2 gd d gq q REF

l

UU i U i UR

+ =

Considering unity power factor, the desired currents are as follows:

*

* 0 0

02

3

d

REFq

gq l

iU UiU R

= =

Control structure

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Fig2 Observer-based control structure

Observer-based control structure is shown is fig2.

Super-Twisting sliding mode

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Robustness property with respect to perturbations and parametric uncertainties.

Smoother than the classic sliding mode control

Advantages:

where , , !nx f b∈( , ) ( , )( )

x f x t b x t uy s x

⋅

= +=

Design12( ), ( ) ( ) ( ) ,( , 0)u s s s sign s sign s dtυ υ λ α λ α= − = + >∫

Control Objective: Force to zero.( )s x

The relative degree with respect to is equal to one.( )s x

Step1:Sliding manifolds design

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Design the switching functions:*

*d d d

q q q

s i is i i= −

= −To find a domain in the system space from which any state trajectory converges to the sliding manifold(sd=0,sq=0).

*1

0 02

*3

2 2

gdd d q

d d d

q qgqq q q d

Ur ui i is u fU UL L C uu fU L Lrs i i i u

L L

ω

ω

⋅⋅

⋅ ⋅

− − − = + = + − − +

Step2: Control design

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Super twisting sliding mode controller can be designed:

12( ) ( ) ( ) ,( , 0)e e sign e sign e dtυ λ α λ α= + >∫

1

20

3

2 32

dT

q

ssLs CsU

s

=

1 1

2 2

3 3

- (s )- (s )- (s )

uuu

υυυ

= The transformed vector s should be designed such that, under the above controller, sd and sq will vanish in finite time.

Super-Twisting Observation Design

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Assuming that only the output voltage(U0) is measured, a super-twisting sliding observer is constructed as a copy of the original system.

Define observation error:

1

2

3 0 0

d d

q q

e i i

e i i

e U U

∧

∧

∧

= −

= − = −

01 3

02 3

0 03 3

( )2

( )2

3 ( ) ( )4

gddd q d

q gqq d q

d d q ql

Ud i Ur i i u k edt L L L

d i UUr i i u k edt L L L

d U U i u i u k edt R C C

ω υ

ω υ

υ

∧ ∧∧ ∧

∧ ∧∧ ∧

∧ ∧∧ ∧

= − + − + −

= − − − + − = − + + −

12

3 3 3 3( ) ( ) ( ) ,( , 0)e e sign e sign e dtυ λ α λ α= + >∫Where

Super-Twisting Observation Design

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The estimation error dynamics are:

11 2 3 1 3

22 1 3 2 3

33 1 2 3 3

( )2

( )2

1 3 ( ) ( )4

d

q

d ql

ude r e e e k edt L L

ude r e e e k edt L Lde e u e u e k edt R C C

ω υ

ω υ

υ

= − + − − = − − − − = − + + −

The sliding surface defined as s=e3=0.

Choosing a large positive constant k3 can assure the convergence of s(e3=0) in finite time.

Observer convergence proof

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Dynamics on the sliding manifold

The equivalent switching function is: 3 1 23

1( ) ( )2 d qe u e u e

k Cυ = +

Substitute into the two equations of error dynamics,

As A is a Hurwitz matrix, there exists a unique positive definite symmetric matrix P that satisfies the equation with positive definite symmetric matrix Q.

TPA A P Q+ = −The Lyapunov function is given as:

12 12

~1 1 11 1 1 1

2 2 2 22 232

( )

1 , , ,0 0 0 02d q d q

dq

e e

r re e ek k k ku u u uL LA A U

k k k kr e e rk CeL Lψ

ω ω

ω ω

⋅

⋅

− − = − = = = − − − −

12 12 12( ) TV e e Pe=

Observer convergence proof

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The derivative of V(e12) is 12 12 12

12 12

( ) ( )V VV e Ae ee e

ψ⋅ ∂ ∂

= +∂ ∂

The first term: 212 12 12 12 12 min 12 2

12

( ) ( )T T TV Ae e PA A P e e Qe Q ee

λ∂

= + = − ≤ −∂

The second term can be expressed as:

12 12 12 12 1222 2 2212 2

( ) 2 ( ) 2 ( )TV e e P e P e eeψ ψ ψ∂

≤ ≤∂

~

12 12 122 2 2223 3

1 2( ) ,2 2dqe A U e e

k C k Cηψ ρ ρ≤ ≤ =

112~ ~ ~

2 2 2max 1 2

2

( ) 2( )T

A A A k kη λ

= = = +

2 212 min 12 max 122 2

( ) ( ) 2 ( )V e Q e P eλ ρλ⋅

⇒ ≤ − +

If ,the origin is globally exponentially stable.min

max

( )2 ( )

QP

λρλ

<

Observation of Source Phase Voltage

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A link current sensor is used to estimate the source phase voltage.

Design the sliding mode observer:

011 1 2 3

12

1 1

1(2 ) ( ),6

( ) ( ) ( ) , link

Ud i r i u u u M edt L L L

e e sign e sign e dt e u i i

υ

υ λ α

∧∧

∧

= − − − − +

= + = − ∫

1 ( )gU M ere eL L

υ⋅ −⇒ = − +

Choose sufficiently large observer gain M, the sliding mode will be enforced in finite time.

1 ( )gU M eυ∧

=

1

2

3

link

ii i

i

=

1 2 3u u u≠ =

2 1 3u u u≠ =

3 1 2u u u≠ =

if

if

if

Observation of Load Resistance

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Error dynamics:0 01 1 2 2 3 3

0 01 1 2 2 3 3

12

0 0

1 ( )2

1 ( ) ( )2

( ) ( ) ( ) ,

l

l

dU U u i u i u idt R C C

d U U u i u i u i k edt CR C

e e sign e sign e dt e U U

υ

υ λ α

∧ ∧

∧

∧

= − + + + = − + + + + = + = −

∫

0 0( ) ( )l

l

U Ue k eR C R C

υ∧

⋅

∧= − − −

Choose sufficiently large observer gain k, sliding mode will be enforced.

0

0

1 1 1( ) ( ) 1 ( )ll

ll

Ue R kC ekC RRR U

υ υ∧

∧= − ⇒ =+

Power Factor Calculation

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1 cos( )h dIPF PF PFI

φ= × = ×

hPF harmonic distortion

φ phase shift between input current and main voltage

1I main current harmonic

I total current RMS

0

1( ( )) ( )T

RMS i t i dT

τ τ= ∫ T is the period of i(t)

Remark: Using Fourier analysis, harmonic distortion measurement, trigonometric modules in matlab, the power factor of the system can easily be obtained.

Simulation Results

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Parameter Value Description

R Load Resistance Ω

r 0.02 Parasitic phase resistance Ω

L 2e-3 Phase inductor H

C 100e-6 Output capacitance F

E 150 Amplitude of source voltage V

f Source voltage frequency Hz

U0ref 650 Desired output DC voltage V; V(0)=1V

1.540 30

t s=

→

1.075 150

t s=

→

Load resistance and frequency are varied to test controller’s ability to handle with varying load and frequency.

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Figure1 Input current and source voltagein case of super-twisting SMC

Traditional SMC results in higher harmonics compared with super-twisting SMC.

Figure2 Input current and source voltagein case of traditional SMC

Input current in phase with source voltage

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Figure3 Source voltage Ug1 estimation within the windows when ilink is equal to i1

Source voltage can be reconstructed with super-twisting observer without designing a low-pass filter.

Source Voltage Estimation

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Figure4 Load resistance R estimator performance

Figure4 shows the performance of load resistance estimation.

Load Resistance Estimation

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Figure5 Output voltage estimatorperformance

Figure6 Current estimator performance0U

∧qi

∧

The performance of observer

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Figure5-7 show the performance of the observer.Figure7 Phase Current estimator performanceai

∧

The performance of observer

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Figure8 Output voltage performance with super-twisting control

A bit higher than the desired voltage level and more oscillating compared with super-twisting SMC.

Figure9 Output voltage performance with traditional sliding mode control

The performance of output voltage

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Figure10 Power factor with super-twisting control

Less value and more oscillations compared with super-twisting control. The super-twisting control was able to produce a power factor that was more than 99%, and can withstand the changing conditions.

Figure11 Power factor with Traditional SMC

Power Factor

Conclusions

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The proposed control method can achieve a power factor close to unity.

Power Balance Condition is taken into account to achieve the desired performance of the system.

The proposed super-twisting observer demonstrates its robustness to the change in operational conditions.

Source Voltage Estimation is achieved with ilink via super-twisting method without using low-pass filter.

Thanks!If you have any questions, I would be pleased to answer them!

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