dynamic phasors in modeling, analysis and control of energy processing systems · ·...

Energy Processing Laboratory, NEU

Dynamic Phasors inModeling, Analysis and Control of Energy Processing

Systems

6/20/02

Alex M. Stankovic

Northeastern University, Boston, MA

1

Research Program Overview


My research program focuses on the interface of control and energy processing.

Emphasis on: 1. dynamical modeling and 2. experimental verification.

Emerging energy conversion technologies: 1. efficiency driven and 2. require closed–loop control - nonlinearity and uncertainty.

NEU Energy Processing Laboratory (1994) is a confluence of research and educa-tional efforts:1. Areas: power electronics, electric drives and power systems,2. Graduate students,3. Sponsors in government and industry,4. Technical collaborations.

2

Research Program Overview


(ONR/VTB)I&I Control

Hilbert Space for Q comp. (NSF)

CAD Methods(EPRI)

Randomized Modulation(NSF, RIA)

ANN in ID(NSF)

IM Control

of DrivesControl

(ARO)

Adaptive

Phasor Dynamics (ONR YIP)

Converters

GM Hughes

Drives(ONR, PEBB)

Resonant

Suppression(NSF, Career)

Systems

Naval

ProbabilisticLoad

Modeling

Power DrivesElectric

ElectronicsPower

Oscillation

Mod

elin

g C

ontro

l(w

ith E

xper

imen

ts)

QuietPMSM

(Draper, Satcon)

3

Presentation Overview


� Background and definitions,

� POWER SYSTEMS - Flexible AC Transmission Systems.

� ELECTRIC DRIVES - AC machine modeling,

� An interlude - extension to polyphase systems,

� POWER ELECTRONICS - active filters and switched-mode DC/DC converters.

4

Background


New challenges in energy processing (power electronics, electric drives, power sys-tems):

� reliance on switching operation for efficiency,

� new dynamic couplings,

� increased performance specifications,

� new problems (e.g., active filtering, pulsed power).

Analytical tools for addressing (“close-to” periodic) system operation:

� “sinusoidal quasi-steady-state” approximation in drives and power systems,

� time-domain simulations (in power electronics and drives),

� systematic exploration of the “middle ground” is largely missing.

5

Background


Main features of dynamic phasors:� large signal models,

� nonlinear,

� physical intuition used for simplifications,

� appealing mathematical structure.

Origins and ideas related to our work:

� power electronics,

� power engineering (“space vectors”, “spiral vectors”, polyphasors),

� nonlinear oscillations (classical averaging and recent variants),

� signal processing.

6

Definitions


A (possibly complex) waveform � �� can be represented on the interval ��

using (short-time) Fourier series:

� ��

��

� � ��

where � � � � � are the complex, slowly time-varying Fourier coefficients,or dynamic phasors.

� � ��

��

� � ��

Our dynamical models describe evolution of � � �� ; for real � �� we have � � � � � � �

7

Definitions


Two useful facts:

Derivative of the � -th dynamic phasor:

� � ��

�

��

� ��

� � � � � �

Multiplication in time domain:

� � � � ��

�

� � � � � � � � � �

8

A Simple Example


Consider a simple RL circuit (R=1, L=0.1) with a �� excitation (V=10), at 60Hzand some initial condition ( � �� A):

0.05 0.1 0.15 0.2 0.25

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Time(s)

Cur

rent

(A

)

9

A Simple Example, cont. 1


The dynamic phasors (according to our definition)

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22

−0.18

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

Fundamental − real part

DC component

Fundamental − imag. part

10

Presentation Map


� Definitions,

� Power systems - Flexible AC Transmission Systems.

� Electric drives - AC machine modeling,


� Power electronics - active filters and switched-mode DC/DC converters.

11

Flexible AC Transmission


Thyristor Controlled Series Capacitors (TCSCs) are finding increasing applicationin power systems.

C

L

+_

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

α τ

φ

[s]

σ

12



Fast and accurate models are needed for simulation and control.

Analytical difficulties stem from the nature of TCSC that amalgamates continuous-time dynamics with discrete events (thyristor firings).

Sampled-data models were the first to offer the needed accuracy, but

� model structure has no clear relation to the system configuration,

� hard to interface with the rest of the system which is usually described withphasor-based continuous-time models.

13



A state-space model for basic TCSC configuration:

��

� � � � �

� �

��

� � � � �

where � is a� � � switching function.

Evaluating the 1-phasor on both sides of each equation (and assuming � � is sinu-soidal) we obtain a 2nd-order (complex) phasor model:

��

� � � � �

� � � ��

��

� � � � � � � � ��

where � � � � � is:

� � � � ��

�

� � ��

14



� � has fast dynamics compared to � � , so we assume

� ��

� �

� � � � � � � ��

yielding

��

� � � � ��

�

� � � � � � � � � � ��

� � � � � � � ��

where� is the prevailing conduction angle

� � � � � � � � � � � � � � � � � � � � � ��

and� � is the reference.

� � � � is computed from steady-state assuming sinusoidal line current � � (in contrastto the conventional approach which assumes� to be sinusoidal).

15



Line current for �� step changes of the firing angle:

0.4 0.5 0.6 0.7 0.8 0.9 10.98

0.985

0.99

0.995

1

1.005

1.01

1.015

1.02

1.025

1.03

[s]0.4 0.5 0.6 0.7 0.8 0.9 1

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

[s]

16

Subsynchronous Resonance


First IEEE benchmark test with TCSC:

TCSC

InfiniteBus

Generator

A B

L LCR

L

s1 s2s

17



0 5 10 15 20 25 30 35 40 45−0.5

0

0.5

1

Conduction angle [degrees]

Mod

al d

ampi

ng (p

er s

econ

d)

TM0

TM2

TM4

TM3

TM1

TM0

TM2

TM4

TM3

TM1

18



0 20 40 60 80 100−15

−10

−5

0

5

10

15

time [s]

Torque deviation [pu]

Shaft Torque − LPB−GEN

19



Expanded view (between 80s and 81s) of torque variations

80 80.1 80.2 80.3 80.4 80.5 80.6 80.7 80.8 80.9 81−5

0

5Shaft torque at different shaft positions

HP

−IP

80 80.1 80.2 80.3 80.4 80.5 80.6 80.7 80.8 80.9 81−5

0

5

IP−L

PA

80 80.1 80.2 80.3 80.4 80.5 80.6 80.7 80.8 80.9 81−10

0

10

LPA

−LP

B

80 80.1 80.2 80.3 80.4 80.5 80.6 80.7 80.8 80.9 81−10

0

10

LPB

−GE

N

80 80.1 80.2 80.3 80.4 80.5 80.6 80.7 80.8 80.9 81−5

0

5

GE

N−E

XC

time [s]

20

Presentation Map


� Definitions,





21

Dynamic Phasors and Space Vectors


Assuming now that all phase quantities are real; let � � � � � � /� :

��

� � � � ��

This complex (scalar) quantity can encode two-dimensional information; for exam-ple, � �� .

� � ��

� � ��

22

Electrical Machines


Space vector model of a three-phase induction machine:��

� ��

� � ��

� � � � ��

� ��

� �

� � � ��

� ��

� � � ��

� � ��

� � ��

��

��

��

��

� � � � � � ��

� ��

� ��

� ��

� � � � � � �

23

Electrical Machines


Dynamic phasor model of a three-phase induction machine:� � � � � � � � � � � � � ��

� � � � � � � � � � � � � � ��

� � � � �

� � � � � � � � � � ��

� � � � � � ��

� � � � � ��

��

��

��

��

� � � � � � � ��

� � � ��

� � � ��

� � � � � � � � � � � ��

� � � ��

� � � ��

��

��

��

��

� � � � � � � � � � � �

��

� � � � � � � ��

� � � � � � ��

��

� � � ��

� � � � � � � � � � � � � � � � � � � � � ��

24

Electrical Machines


Experimental evaluation:

1.67 1.68 1.69 1.7 1.71 1.72 1.73 1.74 1.75 1.76−40

−30

−20

−10

0

10

20

30

40

t (s)

v bc a

nd v

ca (V

)v

ca

vbc

25

Electrical Machines


Experimental evaluation, cont.

1.67 1.68 1.69 1.7 1.71 1.72 1.73 1.74 1.75 1.76−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

ibs

ias

t (s)

i as a

nd i bs

(A

)

26

Electrical Machines


Steady-state equivalent circuit ( � � � �� and � � � � �� ):

rrs

I p, s

L l s

L m

L l r

Vpt

I p r

Vn*tmp

L l s

L m

L l r

In s

I n r

Vn t

Vpt*

( )ωs ( )ωs

+

V

-

rs

+

-

p

+

V

-

rs

+

-

n

rr

2-s

mn+ +

27

Presentation Map


� Definitions,





28

Extension to Polyphase Systems


Dynamical symmetric components - recall � � � � � �� ,

��

� ��

� ��

��

��

��

��

� � �

��

� ��

��

��

� �

� �

� ��

��

��

� �

� �

� ��

��

��

��

� ��

� ��

� ��

��

��

� � � �

� � � ��

��

��

��

� �

� �

� ��

��

�� /� � � � ��

�� /� � � � ��

�� /� � � � ��

��

��

��

� �

� �

� ��

��

29

Properties of Dynamic Symmetric Components


For real waveforms:� ��

Connection with space vectors:

��

� ��

��

30

Asymmetric Faults in Power Systems


Infinite

G

F bus(a)

31



Line voltages

1 1.02 1.04 1.06 1.08 1.1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

v abcs

(p

.u.)

t (s)

vas

vbs

vcs

32



Field current

1 1.02 1.04 1.06 1.08 1.10

0.5

1

1.5

2

2.5

3i fd

(p

.u.)

t (s)

a

b

33

Presentation Map


� Definitions,




� Power electronics - model reduction for switched-mode DC/DC converters.

34

Model Reduction - DC/DC Converters


v (t)in

L

S

D+vi RC

Model in continuous conduction ( � � � � � � when S closed):

��

� � � � � � � � � � � � � ��

��

� � � � � � � � � � � ��

� ��

35



��

� � � � � � � � � � � � � � � � � � � � � � ��

��

� � � � � � � � � � � � � � � � � � � ��

� � � � ��

��

� � � � � � � � � ��

��

� � � � � � � � � ��

��

� � � � � � � � � ��

�

��

�

� � � � � � � � � ��

��

36



0 0.1 0.2 0.3 0.4 0.5 0.60

2

4

6

8

10

12

14

time (ms)

Cap

acito

r Vol

tage

(V)

switched index-0 + index-1 (MFA)

index-0 (MFA)

SSA

37



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

time (ms)

Cap

acito

r Vol

tage

(V)

switched index-0 (CMFA) SSA

index-0 (MFA)

38

Estimation for our Simple Example


Again we consider a simple RL circuit with �� excitation:

��

��

� ��

��

��

�� /� � �

� � � /� � �

� � � �

� � /��

��

��

��

��

� ��

�� /� �

�

��

�

� ��

� ��

��

� ��

��

�

39

Estimation for our Simple Example, cont. 1


True (dash-dotted) and “measured” (solid line) current:

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time(s)

Cur

rent

(A

)

Measured(noisy) current

True current (noise−free)

40



Convergence of estimates:

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

−0.18

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

Dash−dotted − estimates

Solid − true (noise−free)

41



Another view of convergence of the fundamental phasor estimate:

−4 −2 0 2 4 6 8 10

x 10−3

−0.14

−0.138

−0.136

−0.134

−0.132

−0.13

−0.128

−0.126

−0.124Estimated(:) and true (−) phasors trace

Real

Imag

42



Contrast with a model-free (“running FFT”) estimate:

−0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025−0.15

−0.145

−0.14

−0.135

−0.13

−0.125

−0.12

−0.115Measured(:) and true(−) phasors trace

43

Summary


Dynamic phasors yield simple, but powerful large-signal dynamic models.

A unified approach with additional applications in power electronics (resonant con-verters), electric drives (torque ripple minimization) and power systems (protection).

Both refinements and simplifications are possible for various accuracy requirements.

Models are modular, and compatible with engineering experience and intuition,

A timely re-examination of analytical tools in energy processing addresses chal-lenges posed by advances in semiconductor and computer technology.

44

dynamic phasors in modeling, analysis and control of energy processing systems · ·...

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