Contributions to the DC-bus voltage controller of back-to-back voltage source converters

1November, 2005 IECON 2005

Contributions to the DC-bus voltagecontroller of back-to-back voltage source

converters

Electronics Department. Alcalá University.28805 Alcalá de Henares, Madrid, Spain

[email protected]

Santiago Cobreces, Emilio J. Bueno, Felipe Espinosa, Francisco J. Rodríguez, Carlos J. Martín

IECON 2005IECON 2005


TopicsTopics

1. Introduction.

2. DC-bus modelling.

3. DC-bus voltage control design.

independently of RL

in function of RL.

4. Conclusions.


N

nAC

Motor

VSC1 VSC2

Sa2

Sa1

Sa2

Sa1

Sb2

Sb1

Sb2

Sb1

Sc2

Sc1

Sc2

Sa2

Sa1

Sa2

Sa1

Sb2

Sb1

Sb2

Sb1

Sc2

Sc1

Sc2

Sc1

3*L13*L2

3*Co

CDC2

NP

P

CDC1

Da2

Da1

Db2

Db1

Dc2

Dc1

Da2

Da1

Db2

Db1

Dc2

Dc1

ea

eb

ec

PCC

Sc1

TheThe workwork contextcontext

CONDOR Project: “Double converter based on multilevel inverters designed for recovering energy and minimizing electromagnetic emissions”.

VSC 1

Grid

iDC1 iDC2

CDC2

CDC1

VSC 2

AC Motor

uDC measure

uDC control

)(kuDC

)(kuDC∗

Current control

)(kid∗

)(kiq∗

PWM generator

)(* kur

pulses

Grid current measures

Grid voltage measures

SPLL

( ) ( ) ( )tetete cPCCbPCCaPCC ,,

)(kig

r( )kek g

r),(1θ

PWM generator

)(* kur

pulses

Motor measures

( )trω

Current control

Motor control

)(ki ∗r

)(kr∗ω)(kd

∗λ

( )kki rθ),(r

( )krω

Coordination

between controllers

Control of VSC connected to the grid

Control of VSC connected to the AC motor

( ) ( ) ( )tititi cba ,,


Background and objectiveBackground and objective

Previous works[Blasko, et al., 97a], [Hur, et al., 01], [Ottersten, et al., 02] → linear control.[Peña, et al., 01] [Cecati, et al., 03] → non-linear control.[Espinoza, et al., 00a] [Klumpner, et al., 04] →

Stability taking into account the VSC connected to grid. Controller with small CDC.

Problems to solveInclusion of VSC2 current in model and control tasksStability study of back-to-back converters for small (RL·CDC )Evaluation of RL effect in a DC-bus voltage control

Proposals of this workLinearization of the DC-bus model, Design of linear digital controller according to different (RL·CDC) valuesStability analysis in function of RL


TopicsTopics

1. Introduction.



independently of RL

in function of RL.

4. Conclusions.


Models (1/4)Models (1/4)

21 DCDCDC

DCDCLCDCDCqqg iudt

duCuPPiuiePDC

+=+≈≈⋅=

1) DC-bus power balance model

LqqLg

2DCDC PiePPuC

dtd

−=−=2

2) DC-bus storage energy model

iDC1

CDC

iDC2

uDCDC

g

uP

DC

L

uPDCCiGrid side Load side

Dynamic equivalent circuit of the DC-bus

VSC 1 VSC 2

LDC

DC Rui =2Passive Load:

Active Load. VSC1 working as rectifier :

VSC1 working as inverter :

02 >DCi

02 <DCi

LqqDC Piedt

dWC −=21



Disadvantage of model 1:A sensor for iDC2 is needed

Consequently, our design of DC-bus voltage controller is based on model 2.

In our real converter is very difficult to include this sensor.

DC-bus -

DC-bus +

DC-bus

http://www.telec-conference.org/~cobreces/fotos/Condor/img_0583.jpg


Linearization and small signal model from stored energy approach

LqqLgDC

DC PiePPdt

duC −=−=2

21


LqqqqDC PieiedtWdC ~~~~

21

−+=

LP~

qeDCsC

2qi

~W~

LR1

( )1

2

~~

+=

sRCRe

siW

LDC

Lq

q

qi~

qe W~1

2+LDC

LRsC

R

Real Axis-100 -80 -60 -40 -20 0 20 40 60 80 100-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Pole-Zero Map

Imag

inar

y A

xis RL=-50

CDC=200uF RL=50CDC=200uF

RL=50CDC=2000uF

RL=-50CDC=2000uF

RL=-50CDC=1100uF

RL=50CDC=1100uF RL

=∞

RECTIFIER INVERTERPole location in function of CDC and RL

VSC 1 working as rectifier (RL>0): System is stable

VSC 1 working as inverter (RL<0): System is not stable

2DCuW = nq Ee =


Discretization of the linearized small signal model


( )1

2

~~

+=

sRCRe

siW

LDC

Lq

qqi~

qe W~1

2+LDC

LRsC

R

In general, the plant model depends on RL.

12

≈−

DCL

S

CRT

e if , soSDCL TCR 40≥ ( )1

2~~

−≈=

zC

T

eziW DC

S

qq

( )DCL

S

DCL

S

CRT

CRT

Lqq

ez

eReziW

2

2

1~~

−

−

−

−=qi

~W~)(sGZOH qi

~W~)(zG


TopicsTopics

1. Introduction.



independently of RL

in function of RL.

4. Conclusions.


VSC 1

Grid

iDC1 iDC2

CDC2

CDC1

VSC 2

AC Motor

uDC measure

uDC control

)(kuDC

)(kuDC∗

Current control

)(kid∗

)(kiq∗

PWM generator

)(* kur

pulses

Grid current measures

Grid voltage measures

SPLL

( ) ( ) ( )tetete cPCCbPCCaPCC ,,

)(kig

r( )kek g

r),(1θ

PWM generator

)(* kur

pulses

Motor measures

( )trω

Current control

Motor control

)(ki ∗r

)(kr∗ω)(kd

∗λ

( )kki rθ),(r

( )krω

Coordination

between controllers

Control of VSC connected to the grid

Control of VSC connected to the AC motor

( ) ( ) ( )tititi cba ,,

DCDC--bus voltage controllerbus voltage controller (1/3): (1/3): Control objectiveControl objective


Controller parameters:

⎥⎦⎤

⎢⎣⎡ ⎟

⎠⎞⎜

⎝⎛ −−= − 21cos1 ζωζω

SnT

Sq

DCpDC Te

TeCK Sn

⎥⎦⎤

⎢⎣⎡ ⎟

⎠⎞⎜

⎝⎛ −−

−=

−

−

2

2

1cos12

1

ζωα

ζω

ζω

SnT

T

DCTe

eSn

Sn

DCDC--bus voltage controllerbus voltage controller (2/3): (2/3): Controller design (Controller design (RRLL>0 and >0 and CCDCDCRRLL>40>40TTSS))

Controller design

( )1

2−zC

Te

DC

Sq( )kuDC

∗2 ( )kuDC 1−

−z

zK DCpDC

α

(b)

2

1z

(a)

( )kuDC

(c)

1−

−z

zK DCpDC

α

( )kiq∗

totqT ′( )kiq

qe

DCsC2

( )tPL( )tW( )kuDC

∗2

PI

Current controller

ZOH ( )tPg

Plant

( )tiq

( )kuDC∗2

( )12

−zCTe

DC

Sq

The controller design specifications are:10ms < ts < 20ms setting time

ζ ≥ 0.707 damping coefficient

2 2


DCDC--bus voltage controllerbus voltage controller (3/3): (3/3): Simulation results (Simulation results (RRLL>0 and >0 and CCDCDCRRLL>40>40TTSS))

CDC=1000μF, RL=15Ω y TS=200μsCDC·RL > 40 ·TS αDC=0.916 y KpDC=0.0011

L-filter

0 1000 2000 3000 4000 5000 60000

2

4

6

Freq (Hz)

Har

mon

ic i a (A

) 115A (h=1)

0.1 0.15 0.2 0.25 0.3 0.35 0.4650

700

750

800

850

time(s)

u DC

(V)

0.1 0.15 0.2 0.25 0.3 0.35 0.4-200

-100

0

100

200

time(s)

i abc (A

)

uDC* (V)

uDC (V)

LCL-filter

0

0.2 0.25 0.3 0.35 0.4 0.45 0.5650

700

750

800

850

time(s)

u DC

(V)

0.2 0.25 0.3 0.35 0.4 0.45 0.5-200

-100

0

100

200

time(s)i ab

c (A)

uDC* (V)

uDC

0 1000 2000 3000 4000 5000 60000

1

2

3

Freq(Hz)

Har

mon

ic i a (A

) 115 A (h=1)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Root Locus

Real Axis

-40

-20

0

20

40

60

80

G.M.: 21.1 dBFreq: 1.57e+004 rad/secStable loop

Open-Loop Bode Editor (C)

101 102 103 104 105-210

-180

-150

-120

-90P.M.: 60.2 degFreq: 969 rad/sec

Frequency (rad/sec)

Step Response

Time (sec)0 0.002 0.004 0.006 0.008 0.01 0.012

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4


TopicsTopics

1. Introduction.



independently of RL.

in function of RL.

4. Conclusions.


Controller design for Controller design for RRLL>0 and >0 and CCDCDCRRLL<<40<<40TTSS

Block diagram

1−−

zczk 2

1z

)(

)1(1 2

2

DCL

S

DCL

S

CRT

CRT

Lq

ez

eRe−

−

−

−

)(zH

)(zG

)(*2 kuDC

)(ku2DC

With c=0 the system is type 1 Stability for 0<k<kmax

Z= β

αβββ

2251

02+−+−−

<< k where

⎪⎪⎩

⎪⎪⎨

⎧

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

−

−

DCL

S

DCL

S

CRT

CRT

Lq

e

e1

Re

2

2

1

β

α

TS = 200us, eq = 400v, RL = 5Ω, k = 0.1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

50

100

150

Segundos

Respuesta al Impulso del Sistema Realimentado(Ts = 200useg, Cdc = 1000uF, Eq = 400v, R = 5ohm, k = 0.1)

Udc = 650vUdc = 700vUdc = 750vUdc = 800v

CDC < CDC


Controller design for Controller design for RRLL<0 (1/2)<0 (1/2)

With (0<C<1) the system is type 1Stability for kmin<k<kmax

Z= β

2

Block diagram

1−−

zczk 2

1z

)(

)1(1 2

2

DCL

S

DCL

S

CRT

CRT

Lq

ez

eRe−

−

−

−

)(zH

)(zG

)(*2 kuDC

)(ku2DC

)(11

1)( 22

2

2 βα−

=

−

−⎟⎟⎠

⎞⎜⎜⎝

⎛= −

−

zzez

ez

RezG

DCL

S

DCL

S

CRT

CRT

Lq

( )1

z cH z kz−

= ⇒−

1' −−

zCzk

)(1

2 β−zz

)(zH )(zG

)(*2 kuDC )(ku2DC

Equivalent control model

)(1)( 2 β−

=zz

zG

( )1

'1 −

−=

−−

=z

Czkz

CzkzH α

TS = 200us, CDC = 50μF, RL = -100Ω, k = 0.02, c=0.95

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

seconds

Udc = 650vUdc = 700vUdc = 750vUdc = 800v

CDC < CDC


TopicsTopics

1. Introduction.



independently of RL

in function of RL.

4. Conclusions.


ConclusionsConclusions

Two different DC-bus linearized small signal models have been obtained. One of them based on the CDC current, and the another one based on the CDC stored energy.

The direct method in the z-plane has been used to design the digital control solution

If the RLCDC product is bigger enough, the DC-bus voltage controller can be designed independently of RL.The response of this controller, in a back-to-back converter, has been shown for L and LCL grid filters.

If the RLCDC product is not bigger enough, different control solutions have been presented in function of RL, for rectifier and inverter modes of the grid converter.

In future works, adaptive techniques for different loads will be applied to the DC-bus voltage controller.


Thank you for your attention!!!

Santiago Cobreces, Emilio J. Bueno, Felipe Espinosa, Francisco J. Rodríguez, Carlos J. Martín

Department of Electronics. Alcalá University.28871 Alcalá de Henares, Madrid, Spain

[email protected]

IECON 2005IECON 2005

Acknowledgement:This work has been financed by the Spanish administration (CICYT: DPI2002-04555-C04-04).

Contributions to the DC-bus voltage controller of back-to-back voltage source converters

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