Odd-Harmonic Digital Repetitive Control and its application to Active filters control NTU Nayang, February 15h 2006 Ramon Costa Castelló Advanced Control

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NTU Nayang, February 15h 2006

Ramon Costa CastellóAdvanced Control of Energy Systems (ACES)

Institut d’Organització i Control (IOC)Universitat Politècnica de Catalunya (UPC)

Barcelona, Spain

Odd-Harmonic Digital Repetitive Control and its

application to Activefilters control

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Contents• Repetitive Control Basics

– Introduction– Periodic Signals – Discrete Time– Control Scheme– The Odd-Harmonic case

• The active filter application– Introduction– Basic Concept– Single phase– Control Problem– Experimental Setup– Experimental Results

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Introduction• A key topic in classical control theory is the

Internal Model Principle (IMP). – B. Francis and W. Wonham, “Internal Model Principle

in control theory,” Automatica, vol. 12, pp. 457–465, 1976.

• This principle states that if a certain signal must be tracked or rejected without steady-state error, the generator must be inside the control loop, in the controller, or in the plant itself.

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Introduction : Type Concept

• Standard classical control subjects include the IMP concept implicitly when they introduce the system-type concept.

• The type concept can only be applied to polynomial signals (step, ramp, and parabola) whose generator has the form in the Laplace domain.

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Introduction : Type Concept (II)

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Introduction : Systems with periodical disturbances or

references• In practice, many real systems have to

handle tracking and rejecting periodic signals.

Magnet power supply for a proton synchrotron (Nakano and others)

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Introduction : Systems with periodical disturbances or

references (II)

VirtualLaboratory

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Introduction : Power Electronics

• Inverter : Generating a 50/60 Hz signal from dc one (Tracking a reference signal)

• Active filter : Compensation of harmonic signals (Rejecting periodic signals)

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Periodical Signals

• Any periodical signal can be written as:

• The control loop should include:

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Periodical Signals : Generator

Yamamoto, Y. (1993). Learning control and related problems in infinite-dimensional systems. In: Proceedings of the 1993 European Control Conference. pp. 191-222.

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Periodical Signals : Generator I

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Periodical Signals : Generator II

+pT

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Periodical Signals : Generator III

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Digital Case

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Digital Case II

Nz

+

p sT N T

1

1NR z

z

2 2j iN

i ip

z e iN T

pT

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Control Scheme

• Cascade form

• Plug-in Form

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Control Scheme : Cascade form

P(z)

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Control Scheme : Plug-in Approach

pG z cG z

xG z F zNz

Repetitive Controller

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Plug-in Approach : Stability Conditions

1. First stability Condition : The System without the Repetitive Controller must be stable.

2. Second stability Condition

3. Third stability Condition :

cG z

1F z

xG z

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Plug-in Approach : Filter• F(z) should fulfill the second stability condition.• Usually, a low-pass null-phase FIR filter is used.

• To assure unitary gain a DC frequency the parameters must fulfill :

• No causality problems exist because that the filter is in cascade with a N periods delay.

• The filter reduces the open-loop gain at those frequencies at which uncertainty exists (robustness). Unfortunately it slightly moves the open-loop pole positions in z-plane (precision loose).

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Plug-in Approach : Gx(z)

• A common approach to design Gx(z) is

• Unfortunately, this approach cannot be applied to non-minimum-phase plants. Another approach is to cancel minimum-phase zeros and compensate the phase for the non minimum-phase ones:

• kr is fixed by a trade-off between robustness and transient response.

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Odd-Harmonic Case

N

Repetitive controller is order N because it needs to “learn” the N samples defining one period.

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Odd-Harmonic Case (I)

• A function f is odd if the graph of f is symmetric with respect to the origin. Algebraically, f is odd if and only if f(-x) = -f(x) for all x in the domain of f.

• Most signals in power electronics are odd symmetric.

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Odd-Harmonic Case (II)

N

The second semi period can be obtained form the firstone

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Odd-Harmonic Case (III)

Digital repetitive plug-in controller for odd-harmonic periodic references and disturbances Robert Griñó and Ramon Costa-Castelló. Automatica. Volume 41, Issue 1,Pages 153-157(January 2005)

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Odd-Harmonic Case (IV)

Pole-Zero Map

Real Axis

Imag

inar

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xis

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

N=3 odd harmonic

N=3 traditional

Demo

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Odd-Harmonic Case (V)

• Less order (less delay). The closed loop system is faster.

• Robustness– Gain is only introduced where needed– Waterbed

• Many industrial systems include a transformer (a derivator in their open-loop transfer function). Traditional RC many not be internally stable.

• Cannot compensate even harmonics !!!

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Contents• Repetitive Control Basics √

– Introduction– Periodic Signals – Discrete Time– Control Scheme– The Odd-Harmonic case

• The active filter application– Introduction– Basic Concept– Single phase– Control Problem– Experimental Setup– Experimental Results

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Introduction

• Proliferation of nonlinear loads ->This fact has deteriorated the power quality of electrical power systems.

• More stringent requirements proposals IEC-61000-3-{2,4} and IEEE-519.

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Basic Concepts

sv

si Linear Load

Nonlinear Load

Active Filter

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Architecture : Complete Picture (Single Phase)

Full BridgeBoost Converter

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Control Problem: Control Goals

• Current in phase with the voltage waveform:

• Constant average value of the voltage at the DC bus capacitor:

*

2 0 dx V

* sins d ri I t

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Architecture : Boost Converter

1 2 1

2 1

1 sL x x r x vu

C x x

1 2 1

2 1

1 sL x x r x vu

C x x

r L

CsV 2x

1x

r L

CsV 2x

1x

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Architecture : Boost Converter II

1 2 1

2 1

1 sL x x r x vu

C x x

1 2 1

2 1

1 sL x x r x vu

C x x

1 2 1

2 1

sL x ux r x v

C x ux

1 2 1

2 1

sL x u x r x v

C x u x

[ 1,1]u

The averaged model

1,1u

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Control Problem: Current Control loop

1 2 1 sL x u x r x v

2 dx V

1 1d sL x u V r x V

1

1

d

p

Vx s r G s

Lu s sr

1 ( )( ) 1 pp

G sG z z Z

s

ZOH, T

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Control Problem: Voltage Loop

2 1C x u x 1 2C y u x x

22

2

xy

11

sin cos sind r l r l rl

Iload

x t I t a l t b l t

Current loop in steady state

21 2 21 12

k T k kd l l dkT

l

E TC y r I b a b I b

1

12k T k

dkT

E TC y I b

r=0

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Control Problem: Voltage Loop

21

E T

z

1b

PI

2

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Control Problem: Proposed Scheme

• Two control loops :– Current loop : Digital Repetitive Control– Voltage loop : Classical PI Control

sin r t

Boost Converter

Repetitive Controller

PI Controller2x

dVdI

1x lI

*sI

Odd-Harmonic Digital Repetitive Control of a Single-Phase Current Active Filter . Ramon Costa-Castelló, Robert Griñó & Enric Fossas IEEE Transactions on Power Electronics. Volume: 19, Issue: 4, Year: July 2004. E.Page(s):1060- 1068.

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Experimental Setup• Active filter parameters:

– Capacitor: 6600 uF, 450 V DC – Inductor: 0.8 mH– parasitic resistance: 0.04 Ohm– IGBT: 1200 V, 100 A

• Feedback paths (sensors):– Network voltage: voltage

transformer (220V/15V) – Network current: Hall-effect

sensor (TECSA-HA-050053) (50A)

– DC bus voltage: AD-215BY isolation amplifier

• Control hardware:– ADSP-21161 floating-point

DSP – ADMC-200 coprocessor:

A/D channels and PWM generation

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Experimental Results: Nonlinear Load

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Experimental Results: No-Load

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Experimental Results: Full NL load

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Experimental Results: Full load to No-load

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Experimental Results: No-load to full load

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Architecture : Complete Picture (Three Phase)

Active Filter

AC current sensors

AC voltage sensors

DC voltage sensor

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Architecture : Control Scheme

RC

Signal Analyzer

Mode selection

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Architecture : Working Modes

• Branch Equilibration (all net branches have the same active current)

• Harmonic Compensation

• Reactive Compensation

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Experimental Setup : General view

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Experimental setup : IGBT drivers

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Experimental setup : Control hardware

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Experimental setup

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Experimental Results

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Experimental Results: Load Current

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Experimental Results: No Load Current

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Experimental Results: Complete compensation

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Experimental Results: Complete compensation

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Thank you very much

[email protected]

http://www.ioc.upc.es/usuaris/ramoncosta

http://aces.upc.es

Documents

Odd-Harmonic Digital Repetitive Control and its application to Active filters control NTU Nayang, February 15h 2006 Ramon Costa Castelló Advanced Control