analysis of complex behavior of power electronics and...

1TTHEHE HHONGONG K KONGONG

PPOLYTECHNICOLYTECHNIC U UNIVERSITYNIVERSITYChi K. Michael Tse

Department of Electronic& Information Engineering

Analysis of Complex Behavior ofPower Electronics and Applications

Prof. Chi K. (Michael) TseApplied Nonlinear Circuits and Systems Research Group

Hong Kong Polytechnic University, Hong Konghttp://chaos.eie.polyu.edu.hk




Purposes of Today’s Lecture

• To introduce the rich nonlinear behavior of powerelectronics circuits

- an overview of latest research status

• To summarize the main practically-relevantbifurcation scenarios observed in power electronics

- a quick tour of the essentials

• To point out (using an example) how bifurcationanalysis can be used to help design

- a glimpse at practical applications




What are power electronics circuits?




What are power electronics circuits?

The heart isPOWER CONVERTER




Simple dc/dc converters

+–Vin

+Vo–

e.g., buck (step-down) converter





+–+–Vin

+Vo–






+–+–+– 0VVin

+Vo–






+–+–+– 0VVin

+Vo–

• The switch is turned on and off at high frequency.• The output is equal to

†

Vo = DVin

D = duty cycle =

†

on timeperiod





Power supply

+–

+Vo–




Power supply

+–

+Vo–

Vramp

vcon

comp +–

Zf

–+

Vref

FEEDBACK CONTROL




Power supply

+–

+Vo–

Vrampvcon

Vramp

vcon

comp +–

Zf

–+

Vref

FEEDBACK CONTROL




Power supply

+–

+Vo–

Vrampvcon

Vramp

vcon

comp +–

Zf

–+

Vref

FEEDBACK CONTROL

100VAC

AC/DC adaptor




Basic converters

• Buck +–




Basic converters

• Buck +–

complementaryswitches




Basic converters

• Buck +–


• Boost




Basic converters

• Buck +–


• Boost• Buck-boost




Nature of operation

Time varying — different systems at different times

AND

Nonlinear — the time durations are related nonlinearlywith the output function




Nonlinearity – the default property of powerelectronics circuits

• Power electronics engineers/researchers aredealing with nonlinear problems

• Much of power electronics is about identifyingnonlinear phenomena and “taming” them to douseful applications

• Classic examples:• Averaging (R. David Middlebrook, Richard Bass)• Discrete-time modeling (Harry Owen, Fred Lee)• Stability analysis (George Verghese)• Phase-plane analysis/control (Fred Lee, Ramesh Oruganti)• Series approximation (Richard Tymerski)




Classic example of nonlinear study

Boost converter with current-mode control

+–Vin

R

S

Q–+

clock

C R v o

+

–

DL

iL

Iref

Iref

iL

Iref

iL

D < 0.5

D > 0.5

Simple analysisreveals a change ofstability status at acritical duty cycle of0.5.

The circuit isactually ‘stable’beyond the criticalpoint, thoughoperates with alonger period.

This period-doublingphenomenon wasobserved long ago.




Recent study from a bifurcation perspective

Iref

iL

T/CR = 0.125

T/CR = 0.625

sampled

iL

sampled

Iref

With the help of computers, wecan study the phenomenon inmore detail.

Bifurcation diagrams(panaromic view of stabilitystatus)

We can plot bifurcation diagramsfor different sets of parameters

Sampled variable at steady stateversus parameter,e.g., iL(nT) vs. Iref





Iref

iL

T/CR = 0.125

T/CR = 0.625

sampled

iL

sampled

Iref





normal period-1 operation





Iref

iL

T/CR = 0.125

T/CR = 0.625

sampled

iL

sampled

Iref





normal period-1 operation

bifurcation point




Circuits whose bifurcation behaviors have been studiedin detail

Voltage-mode and current-mode controlled simplebuck and boost converters, and many others

+–Vin

R

S

Q–+

clock

C R v o

+

–

DL

iL

–+

Vref

Z f

+–Vin

–+

C R v o

+

–D

L

–+

Vref

Z f

comp




Vin

iL

iL

vo

Period-doubling and chaos in voltage-mode controlledbuck converter

bifurcation diagram

chaotic attractor

Hamill et al. (1990)

+–Vin

–+

C R v o

+

–D

L

–+

Vref

Z f

comp




Period-doubling and chaos in current-mode controlledboost converter

+–Vin

R

S

Q–+

clock

C R v o

+

–

DL

iL

Iref

bifurcation diagram

Iref

iL(nT)




Hopf Bifurcation in parallel boost converters

normal period-1

limit cycle of long period

quasi-periodic orbit

bifurcation diagram

Converter 1

Converter 2

VrefKv1

Kv2

Ki

m

E

Vref

Voffset

Voffset

vcon1

vcon2

RC

rC

v

i1

i2

+

-

+

+

+

+

+-

-

-

-

-

-




Hopf Bifurcation in free-running Cuk converter´

Poincarésection

period-1limit cycle

quasi -periodic

chaos

Hopfbifurcation




Literature review

• Hamill and Jefferies [1988] - First analysis of bifurcation and chaoticdynamics in a first order PWM voltage-mode controlled converter.

• Deane and Hamill [1990] - Analysis of bifurcation in first order andsecond order PWM buck converters.

• Hamill et al. [1992] - Derivation of an iterative map to analyzebifurcation in a buck converter in continuous mode.

• Deane [1992] - First report on chaotic behaviour in a current-controlledboost converter.

• Tse [1994] - Derivation an iterative map to demonstrate period-doublingcascades in a boost converter in discontinuous mode.




Literature review (cont.)

• Chakrabarty et al. [1995] / Fossas and Olivar [1996] - Further study of chaosin a PWM buck converter.

• Dobson [1995] - Study of bifurcation in thyristor and diode circuits

• Poddar et al. [1995] / Batlle et al. [1996] - Control of chaos in dc/dcconverters.

• Tse and Chan [1995] - Study of bifurcation and chaos in a fourth ordercurrent-controlled Cuk converter.

• Chan and Tse [1996] - Study of bifurcation in current-mode converters withand without feedback

• Banerjee et al. [1997] - Analysis of coexisting attractors in buck converter

• Banerjee et al. [1997] - Examination of current-mode converters in the light of“border collision bifurcation”.





• Tse [1997] – Analysis of autonomous Cuk converters using averaged models.

• Di Bernardo et al. [1998] - Study of various sampling and their applications in theidentification of bifurcation and chaos.

• Chan and Tse [1998] – Proof of period-doubling in discontinuous converters usingSchwarzian derivatives

• Di Bernardo et al. [1998] - Analysis of the non-smooth dynamics (such as grazing,skipping and sliding) of dc/dc converters.

• El Aroudi et al. [1999] - Identification of quasi-periodicity and chaos in a boostconverter.

• Mazumder, Nayfeh and Borojevich [1999] - Fast- and slows-scale instabilities.

• Iu and Tse [2000] - Study of bifurcation in parallel converters

• Orabi and Ninomiya [2002] / Tse [2002] - Analysis of power factor correctionconverters





Review articles

D.C. Hamill, Proc. NDES’95, pp. 165-177. 1995.

C.K. Tse, CAS Newsletter, pp. 14-48, March 2000.

I. Nagy, Automatica 42, pp. 117-132, 2001.

S. Banerjee et al., Ch. 1, Nonlinear Phenomena in PE, IEEE Press, 2001.

C.K. Tse and M. di Bernardo, IEEE Proceedings 90, pp. 768–781, 2002.




• Two types of bifurcation* seen in power electronics

• Standard bifurcations (found in other systems as well)• Period-doubling• Hopf (Neimark-Sacker)• Saddle-node

• Border collision (characteristic of power electronics)• Abrupt change of behavior due to a structural change

Current state of findings

*Bifurcation refers to sudden change of qualitative behaviour of a dynamicalsystem when a certain parameter is varied.




• Standard bifurcations• Buck converters (voltage-mode) – period-doubling• Boost converters (voltage-mode) – Hopf• Dc/dc converters in DCM – period-doubling• Most dc/dc converters (current-mode) – period-doubling• Other types – variety: saddle-node, crisis, etc.

• Border collision (characteristic of power electronics)• All standard bifurcations are interrupted by bordercollision

Current state of findings

Who will get what?




• Standard bifurcations• Loss of stability• No structural change• Standard appearance bifurcationdiagrams

Comparisons

• Border collision• Loss of “operation”• Structural change• Non-standard apppearance inbifurcation diagrams, e.g.,bending, jumps, etc

Structural change in switching converters= Alteration in topological sequence

e.g., change of operating mode, reaching a saturation boundary




Border collision –signature bifurcation of power electronics circuits

Border collision

Non-smooth phenomena “always” observed in power electronics circuits

boost converter under current-mode control




Border collision

Essence: standard bifurcation being interrupted

boost converter under current-modecontrol




Border collision – a practical viewpoint

Excessive swing of control signalDuty cycle saturationOut-of-range operation, preventing continuation of standard bifurcations

e.g., buck converterin voltage-modecontrol

+–Vin

–+

C R v o

+

–D

L

–+

Vref

Z f

comp







period-doubling

+–Vin

–+

C R v o

+

–D

L

–+

Vref

Z f

comp







period-doublingborder collision

+–Vin

–+

C R v o

+

–D

L

–+

Vref

Z f

comp




• What models to use?

• What techniques to use?

• An open question:• What are the applications?• (What do we use the results for?)

Basic problems in analysis




What models to use?

• Averaged models

• Discrete-time models

†

dxdt

= f (x,m,t)

averaged behaviour

†

xn+1 = F (xn ,m)

t

t

x(t)

xn

xn+1

simple - details destroyed

complex - accurate




What models to use?Clue: Typical nonlinear behaviour observed in dc/dc converters

• Standard bifurcations

• Hopf bifurcation,

• Period-doubling bifurcation,

T

t

t




What models to use?Clue: Typical nonlinear behaviour observed in dc/dc converters

• Standard bifurcations



2TT

t

t




What models to use?Clue: Typical nonlinear behavior observed in dc/dc converters

• Phase space view



v

i

i

v

limit cyclelow-freq. orbit

2T orbit

v

v

i

i




What models to use?Fast and slow scale dynamics

• Fast and slow scales











2T orbit

v

v

i

i









2T orbit

v

v

i

i

low-frequencyphenomenon









2T orbit

v

v

i

i


– averaged modelsare capable









2T orbit

v

v

i

i



high-frequencyphenomenon









2T orbit

v

v

i

i



high-frequencyphenomenon

– discrete-timemodels are needed




What models to use?Clue: Experiments and computer simulations

e.g., In parallel boost converters, as feedback gain increases,we observe a series of changes…Stable period-1 quasi-periodic limit cycle





e.g., In a current-mode boost converter, we see period-doubling:

bifurcation diagram

Iref

iL(nT)





EXAMPLES:For the parallel boost converters under master-slave control,we see a slow-scale phenomenon.

Thus, averaged models should be adequate!(Iu and Tse, ISCAS’2002)

For the current-mode controlled boost converter, we see fast-scale phenomenon.

Thus, we must resort to discrete-time models.(Chan and Tse, IEEE TCAS1 1997)




Analysis techniques

The basic questions of practical importance are

Where and when it happens?Location of boundary of operation – bifurcation point

How it happens?Identification of the type of bifurcation




Technique – Averaging approach

Averaging approach• Derive a set of continuousaveraged equations:

• Examine the Jacobian, J(XQ) andfind the loci of the eigenvalues whena bifurcation parameter is varied.

• Identify the condition for theeigenvalue(s) moving across theimaginary axis in the complex plane:

• e.g., a pair of complex eigenvaluesmoving across the imaginary axisimplies a Hopf bifurcation.

)(xfx =&

s

jw




Advantages:

• Widely used and well known.

• Relatively easy to derive the continuous averaged equation.

Limitation:

• Only able to predict low-frequency slow-scale bifurcationbehaviour such as Hopf bifurcation.

Technique – Averaging approach




Discrete-time map approach• Derive a discrete time map (iterative function f):

• Examine the Jacobian, J(XQ) and find the loci ofthe eigenvalues when a bifurcation parameter isvaried.

• Identify the condition for the eigenvalue(s)moving out the unit circle in the complex plane:

••e.g., one of the eigenvalues moving out through -1the implies a period-doubling.

†

xn+1 = f (x n ,d)

†

det[lI - J(XQ )] = 0

Technique – Discrete-time map approach




Advantages:

• Provide a relatively complete behavioral information.

• Able to predict standard bifurcations such as period-doublingbifurcation, Hopf bifurcation and saddle-node bifurcation.

Limitation:

• Derivation of the iterative map is more complicated comparedto the continuous-time averaged equation.

Technique – Discrete-time map approach




Basic principle:

It usually involves change of the form of the qualitative model beforeand after the collision.

Results (reported in Physics literature):

There are theoretical publications (C. Grebogi, H.E. Nusse, S. Banerjee,M. di Bernardo) on the type of transition at the collision. Thesetransitions are usually “abrupt” changes, e.g.,

from period-1 to chaosfrom period-1 to another period-1

Analysis of border collision




Mechanism of border collision: circuit viewpoint

The basic distinguishing mechanism is STRUCTURAL CHANGE, as explained in

Chapters 1 and 5 ofC. K. Tse, Complex Behavior of Switching Power Converters, Boca Raton: CRC Press, 2003.

Smooth bifurcation involves no structural change, whereas border collision MUSTinvolve a structural change.

What is a structural change in power electronics?





The basic distinguishing mechanism is STRUCTURAL CHANGE, as explained in

Chapters 1 and 5 ofC. K. Tse, Complex Behavior of Switching Power Converters, Boca Raton: CRC Press, 2003.

Smooth bifurcation involves no structural change, whereas border collision MUSTinvolve a structural change.

What is a structural change in power electronics?

Alteration of topological sequence





Quick glimpse of concept:

Suppose a converter is operating in a certain mode that has the following switchingsequence:

on - off - on - off - on - off - on - off - ….

When a parameter is varied up to a certain point, the switching sequence suddenlyalters to:

on - off - on - on - on - off - on - on - on - off - …

This is a change of topological sequence or a structural change.




Detecting border collision by symbolic analysis

Formal definitions:

Definition 1 — A switching block is a sequence of switch states which is takenwithin one particular switching period.

Definition 2 — A block sequence is a symbolic sequence of switching blocks thatdescribes the way in which the block of switch states changes as time advances.

Applications:

Any solution can be represented by an infinite sequence of switching blocks.

Hence, a periodic solution implies a periodic sequence of switching blocks.However, the converse is not true.




Detecting border collision by symbolic analysis

Notations

Suppose b1, b2, b3, …, bm are the switching blocks.

We denote (b1b2b3…bm)n as a finite block sequence that repeats the block sequence(b1b2b3…bm) n times.

Thus, (b1b2b3…bm)∞ is periodic block sequence repeating (b1b2b3…bm).

Moreover, (∞) is an aperiodic block sequence.




Method for identifying border collision in circuits

Detecting border collision

Basic Idea — Consider a switching power converter with parameter a Œ ¬. Supposethe block sequence for a < ac is B1 and the block sequence for a > ac is B2. Then,border collision occurs at a = ac if B1≠ B2.

aca

B1 B2block sequence block sequence

a < ac a > ac

border collision if B1≠ B2




Example 1: Buck converter with voltage-mode control

Operation: Switch on if vcon < Vramp Switch off otherwise





If there are i switchings per cycle, wewill have i + 1 switch states in aswitching block sequence. We maysimply label the block as

2i+1 if the first state is OFF

2i+2 if the first state is ON

For example, if we consider up to 2switchings per cycle (i=0,1,2), thenwe have up to 6 possible blocksequences.

block 1 block 2

block 3 block 4

block 5 block 6





Circuit parameters:L = 20 mHE = 5VT = 400 µsa = 8.4C = 47 µFVref = 11.3 VVL = 3.8 VVU = 8.2 V

Objective:

Find all bifurcation boundary curves.

- Standard bifurcations: PD

- Border collision: BC





Solid lines : bifurcation boundaries

Regions of different colours:different block sequences

Solid lines separating colouredregions: Border collision

Solid lines within one region:standard bifurcation




Example 2: Boost converter with current-mode control

+–Vin

R

S

Q–+

clock

C R v o

+

–

DL

iL

Iref

Iref

iL

Iref

iL

D < 0.5

D > 0.5

Simple analysisreveals a change ofstability status at acritical duty cycle of0.5.

The circuit isactually ‘stable’beyond the criticalpoint, thoughoperates with alonger period.

This period-doublingphenomenon wasobserved long ago.





Circuit parameters:L = 1.5 mHE = 5VT = 100 µsR = 40 ΩC = 8T/R

Suppose discontinuous mode operation is notconsidered. There are two possible blocks forthis particular converter operation.

0 : on-off

1 : on

Iref

Iref

block 0

block 1





Circuit parameters:L = 1.5 mHE = 5VT = 100 µsR = 40 ΩC = 8T/R

The switching block foundfor Iref= 0.53 A is on-off-on-off…

The block sequence is (0)∞.





Let us increase Iref to 0.55 A andfind the block sequence again.

It is found that the blocksequence is the same as before,i.e., (0)∞. Thus, there is noborder collision.

Then, we find the periodicity,which is different from theprevious case.

Hence, the system hasexperienced a standardbifurcation.





Let us increase Iref further to0.73 A and find the blocksequence again.

It is found that the blocksequence has changed to1000101010….

Hence, the system hasexperienced a border collision.





Bifurcation diagram on theparameter space can begenerated by computer usingsimple symbolic analysis. Here,we have

• Period-doubling • Border collision

Period-doubling

Border collision




Summary of observations

Standard bifurcations such as period-doubling and Hopf bifurcation arecommonly observed in dc/dc converters.

Border collision bifurcation comes intoplay to disrupt the growth of standardbifurcations.




Summary of observations

Standard bifurcations such as period-doubling and Hopf bifurcation arecommonly observed in dc/dc converters.

Border collision bifurcation comes intoplay to disrupt the growth of standardbifurcations.

— practically more important becausestandard bifurcation is always the firstbifurcation next to the usual stable operation

— signature phenomenon in powerelectronics




Possible engineering applications

Prediction / Better understandingSystematic collection of results concerning bifurcation to form auseful design guide.

operation boundaryinstability features —

[ e.g., recent finding on PFC converter ]

DesignUse of chaotic operation to advantage.

transient speed ?EMC ?




Application example– bifurcation analysis in PFC boost converter

PFC boost rectifier

= typically a boost converter undercurrent-programming control

The input current is forced to trackthe input voltage waveshape





PFC boost rectifier

= typically a boost converter undercurrent-programming control

The input current is forced to trackthe input voltage waveshape

Note: essentially slopecompensation





We observe asymmetrical slopecompensation in this PFC boostrectifier: +ve slope compensation in [0, p/2]--> less stable –ve slope compensation in [p/2, p]--> more stable

Bifurcation analysis can revealinteresting phenomenon…

where just for convenience.

Discrete-time map:

†

in +1 =Mc + 1-v /Vin

Mc +1

Ê

Ë Á Á

ˆ

¯ ˜ ˜ in + higher order terms

†

Mc =mcLvin





Bifurcation analysis

Suppose the input voltage is

Relating the compensation slope mc withthe input voltage variation for the PFCcase:

mc = – Question:

Would there be fast-scaleinstability (e.g., period-doubling)?At what phase angle would itoccur?





Bifurcation analysis

Jacobian:

Characteristic multiplier:

Period-doubling occurswhen l = –1.

Critical phase angle (algebra omitted):





Fast scale instability at phase angle





This formulaallows us to do anumber of things:

e.g.,Predicting fast-scale instability





Defining stabilityboundaries





†

0 < qc1 <p

2p

2< qc2 < p

†

qc1 =p

2p

2< qc2 < p

†

qc1 >p

2

qc2 <p

2

region 2 region 3




Conclusion

• Power electronics engineers/researchers are dealing withnonlinear problems

• Much of power electronics is about identifying nonlinearphenomena and “taming” them to do useful applications

Recap:

For power electronics (nonlinear systems in general),

– “stability” refers to operation in the expected regime– a variety of ways the system can become unstable (to get away from the usual operation)– a number of affecting parameters

— BIFURCATION ANALYSIS




Future work

Engineers frequently ask:

What are the applications of chaos and bifurcation studies?

Topics of future research:

• Reorganizing results in terms of practical operatingconditions and parameters• Developing design-oriented bifurcation procedures• Identifying new phenomena in practical circuits




To probe further

Books• C.K. Tse, Complex Behavior of Switching Power Converters, Boca Raton: CRC Press,2003.• S. Banerjee and G.C. Verghese (Eds.), Nonlinear Phenomena in Power Electronics:Attractors, Bifurcations and Nonlinear Control, New York, IEEE Press, April 2001.

Review paper• C.K. Tse and M. di Bernardo, “Complex behavior of switching power converters,” Proceedings ofthe IEEE, vol. 90, no. 5, pp. 768–781, 2002.

Journals• IEEE Transactions on Circuits and Systems Part I• International Journal of Bifurcation and Chaos• International Journal of Circuit Theory and Applications




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analysis of complex behavior of power electronics and...

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