Stability of Power Electronic Systems S.D. Sudhoff Fall …sudhoff/ece695 power...power converters...

Stability of Power Electronic Systems

S.D. Sudhoff

Fall 2010

Some Definitions (Zak, 2003)

• System description (non-autonomous)

( ) 00

))(,f()(

xx

xx

==

t

tttp

• System description (autonomous)

2

( ) 00

))(f()(

xx

xx

==

t

ttp


• Conditions for solution

• Solution is denoted

3

),;()( 00 xxx ttt =


• Equilibrium point

• Translation of equilibrium point

tt e ∀= 0),( xf

4

exxx −=~


• An equilibrium state xe is said to be stable if for any given t0 and any given positive scalar ε their exists a scalar δ (t0,ε) such that

000000 ),;(),()( tttttt ee >∀<−⇒<− εεδ xxxxx

5


• An equilibrium state is said to be convergentor attractive if for any given t0 there is a positive scalar δ 1 (t0)

et

e ttt xxxx →⇒<−∞→

)(lim)()( 000 δ

6

et

e ∞→000


• An equilibrium state is asymptotically stable if stable and convergent

• An equilibrium state is uniformly stable if δ (t0,ε)= δ (ε)

• An equilibrium state is uniform convergent if • An equilibrium state is uniform convergent if δ 1 (t0) = δ 1

• An equilibrium state is uniformly asymptotically stable if is uniformly stable and uniformly convergent

7


• An equilibrium state xe is said to be unstable if there is an ε > 0 so that for any δ > 0 there exists x(t0) such that if

then

δ<− et xx )( 0

then

for some t1 > t0

8

ε≥− et xx )( 0


9


• Lyapunov’s Indirect Method

• Let the origin x=0 be an equilibrium state of the nonlinear system px=f(x). Then the origin is an asymptotically stable equilibrium of the nonlinear system if A, the Jacobianmatrix of nonlinear system if A, the Jacobianmatrix of f, evaluated at the origin, has all its eigenvalues in the open left-half plane.

• Take ECE675 for proof

10

Stability of Power Electronic Systems

• Reasons for concerns

• Subtleties

11

Physical Definition of Bus Stability

• A bus would be defined to be locally dynamically stable at a given operating condition provided that the bus voltage is constant at that operating point, and that if the bus voltage were perturbed it would return to

12

bus voltage were perturbed it would return to its original value

• Bus voltage is either dc, or q- and d-axis voltage in a synchronous reference frame

Practical Physical Def. of Bus Stability

• A bus is defined to be locally dynamically stable at a given operating point provided that the bus voltage variation is restricted to frequencies corresponding to the switching frequencies (and harmonics thereof) of the

13

frequencies (and harmonics thereof) of the power converters in the system

• A bus is said to be regionally dynamically stable over a set of operating points if it satisfies the definition of being locally dynamically stable for all operating points in the set (region).

Physical Definition of System Stability

• System stability is an amalgamation of bus stability; in particular, a system is said to be locally dynamically stable at a given operating condition if all busses in the system are dynamically stable at that operating point

14

dynamically stable at that operating point

• A system is regionally dynamically stable if all busses in the system are regionally dynamically stable at all relevant operating conditions.

Consider a System

IMControls

Vol. Reg./Exciter

iabcsvdci

vdcsvr

ωrm,im

Te,desvdcs*

15

SM IMTurbineMechanical

Load

vdci

3 - Uncontrolled

Rectifier

φ 3 - Fully Controlled

Inverter

φLCFilter

CapacitiveFilter

Tie Line

System Performance

20

-20

ias, A *

16

100 ms

20

550

-20

250

ias, A

vdci, V

A Simple System

17

Linearizing…

18

Equivalent Circuit

19

Transfer Function

20

Characteristic Polynomial

21

A Condition for Stability

22

Another Condition for Stability

23

General Modeling Requirements

• We assume that all components can be represented in the following form

),( uxfx =

dt

d

∫=t

ττ1ˆ

24

dt

),( uxgy =

∫−

=t

Tt

dxT

x ττ )(1ˆ

Model Linearization

• We will be using linear analysis techniques

• At an operating point

),(0 00 uxf= ),( 000 uxy g=

25

• Perturbation from operating point

0uuu −=∆0xxx −=∆ 0yyy −=∆

Model Linearization

• Linearized System Representation

• Matrix Definitions

uBxA∆x ∆+∆=dt

duDxCy ∆+∆=∆

26

00,

),(

uxxuxf

A∂

∂=00,

),(

uxuuxf

B∂

∂=

00,

),(

uxxuxg

C∂

∂=00,

),(

uxuuxg

D∂

∂=

Writing Specification to Address Stability

27

Immittance from Linearized Models

• Going to the frequency domain

DBAIC +−= −1)()( ssI

28

Contour Evaluations of Complex Functions

• Consider the transfer function

13

1)(

2 +++=

ss

ssH

0.5

1

1.5

2

2.5

Closed Contour

Contour Evaluation

29

• A contour evaluation appears to the right

-0.5 0 0.5 1 1.5 2 2.5-2.5

-2

-1.5

-1

-0.5

0

The Cauchy Principal

• A contour evaluation of a complex function will only encircle the origin if the contour contains a singularity of that function

30

Cauchy Principal: Example 1

)1)(1(

5.0)(

+−+=

ss

ssH

0.5

1

1.5

Closed Contour

Contour Evaluation

31

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4-1.5

-1

-0.5

0

Cauchy Principal: Example 2

)1)(1(

5.0)(

+−+=

ss

ssH

0.5

1

1.5

Closed Contour

Contour Evaluation

32

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4-1.5

-1

-0.5

0

The Cauchy Principal Explained

∏

∏

=

=

−

−=

p

z

n

kk

n

jj

ps

zs

asH

1

1

)(

)(

)(

33

∑ ∑= =

−∠−−∠=∠z pn

j

n

jjj pszssH

1 1)()()(

The Cauchy Principal: Zeros

34

The Cauchy Principal: Poles

35

The Cauchy Principal: Conclusions

zpccwe NNN −=

NNN −=

36

pzcwe NNN −=

The Nyquist Stability Criterion

• The Nyquist stability criterion is useful because it allows us to gain insight on the stability of a closed loop plant based on properties of an open loop plant

• Consider the following:

37

• Consider the following:

)(sGΣ+−

yu

Nyquist Stability Criterion (Cont)

)(sGΣ+−

yu

38

Nyquist Stability Criterion (Cont)

39

The Nyquist Contour

∞j

∞

40∞− j

Nyquist Stability Criterion

• Consider a Nyquist evaluation of

• We have

)(1 sG+

pzcwe NNN −=

uclpz NN =

41

• Thus we have …

uclpz

uolpp NN =

uolpcweuclp NNN +=

Nyquist Stability Criterion

Nyquist Stability Criterion: The number of unstable closed loop poles is equal to the number of unstable open loop poles plus the number of clockwise encirclements of -1 by the contour evaluation of the open loop plant (i.e. G(s)) over

42

evaluation of the open loop plant (i.e. G(s)) over the Nyquist contour.

Comments on Path

max|| ω=s

maxωjs =

Point 2

Point 3

Point 4

43

maxωjs −=

maxω=s0=s

Point 1

Point 2 Point 4

s-plane

Consider a Source

• Definition

• Model

44

sTss visZv += )(

00 )0( sssT iZvv −=

Consider a Load

• Definition

• Model

45

lNll ivsYi += )(

00 )0( vYii lllN −=

Properties of Source and Load

)(

)()(

sD

sNsZ

s

ss =

46

)(

)()(

sD

sNsY

l

ll =

Consider a Source-Load System

sZ

lYvsTv lNi

si li

++

--

Source Load

47

Nyquist ImmittanceCriterion

• Consider

)1( lss

lNssTs

YZD

iNvDv

+−=

48

• Thus we have the Nyquist Immittance Criterion: A source – load system is stable provided that the Nyquist evaluation of ZsYl does not encircle -1.

Quick Review – Gain and Phase Margin

49

Stability Criteria

1/GM

ImaginaryAxis

Unit CircleESACCriterion

GMPMCriterion

50

OpposingArgumentCriterion

PM 1/GM

Real Axis

Middlebrook Criterion

MiddlebrookCriterion

• We have

PM 1/GM

ImaginaryAxis

Unit Circle

Real Axis


ESACCriterion

GMPMCriterion

51

• As a design specification

GMZY

sl ||

1|| <


Opposing Argument Criterion

Opposing

PM 1/GM

ImaginaryAxis

Unit Circle

Real Axis


ESACCriterion

GMPMCriterion

52


Gain and Phase Margin Criterion

GMZY

sl ||

1|| <

Opposing

PM 1/GM

ImaginaryAxis

Unit Circle

Real Axis


ESACCriterion

GMPMCriterion

- OR -

53

( )( )PMsZsY

PMsZsY

osl

osl

+−≥∠+∠

−≤∠+∠

∠

∠

180)()(

and180)()(


- OR -

ESAC Criteria

PM 1/GM

ImaginaryAxis

Unit Circle

Real Axis


ESACCriterion

GMPMCriterion

54



Comparison of Stability Criteria

• Parameters+_ C - R

Source Load

Vs

r L

Ω= mr 300

,10mHL =

Ω= 3.24R

55

+_ C - R

Source Load

Vs

r L

(a) component grouping 1

(b) component grouping 2

Ω= 3.24R

mFC 40=mFC 5.0=

Component Grouping A

ESACCriteria

Nyquist Contour,Stable Case

ImaginaryAxis

Arcs at |s| =

3

400π

56

GMPMCriteria

Criteria

Real Axis

Nyquist Contour,Unstable Case

|s| =

3

-3

-3

400π

Component Grouping B

ImaginaryAxis

ESAC Criteria

GMPMCriiteria

1.5

57

Nyquist ContourStable Case

Nyquist ContourUnstable Case

Real Axis

ESAC Criteria

-3

-1.5

Design Specs From Arb. Stab. Crit.

• At a frequency at a point

Nyquist Evaluationof Source Impedance

Zs,a

58

ESAC Stability Criteria

Real Axis

ImaginaryAxis

Zs,a

sbsc

sdas

babl Z

sY

,, =

Design Specs Based on Arb. Stab. Crit.

• At a frequency over all points

Gain

, dB

40

35

30

25

20

59

Gain

, dB

Phase, Degrees

20

15

10

5

0

-5

-1060 70 80 90 100 110 120 130 140

ForbiddenRegion

Design Specs Based on Arb. Stab. Crit.

• Over all frequencies

60

40

20

ma

gn

itude

, d

B

Stability Constraint in Admittance Space

60

20

0

-20

-40300

200

100

00

-2

24

log of frequency, Hzphase, degrees

ma

gn

itude

, d

B

Inflection

Load Admittance,Unstable Case

Load Admittance,Stable Case

Comments

• Conservativeness

• Amenability to Configuration

61

• Simplicity of Formulating Design Specs

Generalized Immittance

• Consider the following system

IMControls

Vol. Reg./Exciter

iabcsvdcsvr

ωrm,im

Te,desvdcs*

62


Load

iabcsvdci

vdcsvr

3 - Uncontrolled

Rectifier


Inverter

φLCFilter

CapacitiveFilter

Tie Line

Source Impedance Vs. Operating Point

• Output Power from 0 to 1 p.u. in 5 steps

20

40

60

mag

nitu

de,

dB

63

-2-1

01

23

-100

0

100-40

-20

0

20

log of frequency, Hzphase, degrees

mag

nitu

de,

dB

Generalized Source Impedance

• Nominal speeds from 228-279 rad/s

• Output voltage references from

64

references from 380-420 V

• Output powers from 0 to 4.07 kW

Generalized Load Admittance

• Speed from 0-200 rad/s

• Torque commands from 0-20.9 Nm

65

0-20.9 Nm

• Input voltage from 380-420 V

Generalized Nyquist Evaluation

• Source Impedance

• Load Admittance

)()( sZsZ G⇒

66

)()( sYsY G⇒

Product of Generalized Quantities

• Consider

)()( sXsx G∈ )()( sYsy G∈

67

Implications on Stability

• Consider Nyquist Evaluation of )()( sYsZ GG

68

An Example

• Consider the followingΩ= 3.97loadR

Ω= 3.24loadR

0.3 Ω 5 mH

69

+_ 500 Fµ - Rload

Source Load

vs

0.3 Ω 5 mH

Normal Nyquist

1

UnstableCase

70

-2 -1

-1

0StableCase

Let’s Add Variation

• Allow all parameters to vary by 1%1

1|)()( ωjs

Gl

Gs sYsZ =

71

-2 -10

-1

Generalized Middlebrook

• Standard Middlebrook

• Generalized MiddlebookGMsYsZ ls

1)()( <

xX = max

72

GXxG

xX∈

= max

GMsYsZG

Gl

G

Gs /1)()( <

Generalized Middlebrook

• In terms of design specs:• If we know source

G

Gs

G

Gl

sZGMsY

)(

1)( <

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• If we know load

G

Gs

G

Gl

sYGMsZ

)(

1)( <

Generalizing Other Criteria

Nyquist Evaluationof NominalSource Impedance

sbsc

PasZ ,

GasZ ,

absZ ,

ε

abs

babl Z

SY

,, =

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ESAC Stability Criteria

Real Axis

ImaginaryAxis

sc

sd

εbt

∫ −P

asZas

abs

bb dssZ

Z

st

,

)(,,

Case Study – Original System

IMControls

Vol. Reg./Exciter ωrm,im

Te,desvdcs*

75


Load

iabcsvdci

vdcsvr

3 - Uncontrolled

Rectifier


Inverter

φLCFilter

CapacitiveFilter

Tie Line

Gen. Adm. Constraint and Admittance

76

Original System Time Domain Perform.

20

20

-20

ias, A *

77

100 ms

20

550

-20

250

ias, A

vdci, V

Let’s Modify the Control

78

Modified Control

79

Adm. Constraint and Modified Adm

80

Modified System Time Domain Perform.

20

20

-20

ias, A *

81

100 ms

20

550

-20

250

ias, A

vdci, V

Stability of Power Electronic Systems S.D. Sudhoff Fall …sudhoff/ece695 power...power converters...

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