AND REPETITIVE CONTROL OF GRID CONNECTED...

H∞ AND REPETITIVE CONTROLOF GRID-CONNECTED INVERTERS

Qing-Chang [email protected]

Electrical Drives, Power and Control Group

Dept. of Electrical Eng. & Electronics

The University of Liverpool

United Kingdom

Q.-C. ZHONG: H∞ AND REPETITIVE CONTROL OF GRID-CONNECTED INVERTERS– p. 1/33

Outline

Introduction

System structure

Modelling and controller design

Experimental results

Overview of other projects


DC-AC converters in the contextof distributed generation

Local generator

Diode

Rectifier

DC-AC

Converter

Micro-grid

grid DC link

Gas turbines Wind-mills etc.

Fuel cells Photo-voltaic etc.


Control problems involved

+

-

Ls , Rs va

vb

vc

ia

ib

ic

ea

eb

ec

VDC

C

vga

vgb

vgc

Circuit Breaker

Lg , Rg

Power quality control: to reduce THD

Provision of a a non-drifting neutral point/line

Power flow control: to regulate P/Q

Phase-locked loop (PLL): to synchronise with the grid


Power quality improvementPower quality is an important problem for renewableenergy and distributed generation. The maximum totalharmonic distortion (THD) of output voltage allowedis 5% (120V −69kV ). The maximum THD allowed incurrent is below:

Odd harmonics Maximum current THD

< 11th < 4%

11th − 15th < 2%

17th − 21th < 1.5%

23rd − 33rd < 0.6%

> 33rd < 0.3%


H∞ repetitive voltage-controlled VSIs

DC power source

Inverter bridge

LC filter

Transformer

PWM modulation

Internal model M and stabilizing compensator C

ia ib ic

Id* Iq*

ua ub uc

-

PI controllers

uref e

Id Iq

abc

dq + +

θ

Power controller

+

Voltage controller

PLL

abc

dq

ugb uga ugc

abc

dq

u Ud Uq

+ +

- +

- +

θ

Id Iq

An inner voltage loop to track the reference voltageuref

An outer power loop to regulate active powerP and

reactive powerQ.Q.-C. ZHONG: H

∞ AND REPETITIVE CONTROL OF GRID-CONNECTED INVERTERS– p. 6/33

Repetitive controlIncorporating an internal model, a local positivefeedback of a delay linee−τds cascaded with alow-pass filterW (s), to track or reject periodicsignals with a periodτ . Normally,τd ≈ τ .

A stabilising controller is normally needed.

uref plant

stabilizing

compensator

u

e ug

sdesW

τ−)(

+

internal model

+ w

p

P

C

M

Q.-C. ZHONG: H


Poles of the internal model

−18 −16 −14 −12 −10 −8 −6 −4 −2 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 10

4

Re

Im

true poles

approximated poles * o

Resk ≈1

τd

lnτdωc

√

(τdωc)2 + (2kπ)2= −

1

2τd

ln

(

1 + (2kπ

τdωc

)2)

,

Im sk ≈2kπ − tan−1 2kπ

τdωc

τd

≈2kπ

τd

(

1 −1

τdωc

)

.


Single-phase circuit

PWM ic Rf Lf Rg

u

grid Cf

ug

filter inductor grid interface inductor uc

Lg i1 i2

Sc

+ - VDC

neutral

uf

Rd

Inverter bridge

uo

Consists of the inverter bridge, an LC filter (Lf

andCf ), a grid interface inductorLg, and acircuit breakerSC

uf ≈ u: the PWM block and the inverter bridgecan be ignored when designing the controller.


Modelling

States:x =[

i1 i2 uc

]T

External inputs: w =[

ug uref

]Tandu.

Output:e = uref−u0, whereu0 = uc+Rd(i1−i2)is the output voltage of the inverter.

The plantP can then be described by the state equation

x = Ax + B1w + B2u

and the output equation

y = e = C1x + D1w + D2u


A =

−Rf+Rd

Lf

Rd

Lf− 1

Lf

Rd

Lg−

Rg+Rd

Lg

1Lg

1Cf

− 1Cf

0

,

B1 =

0 0

− 1Lg

0

0 0

, B2 =

1Lf

0

0

,

C1 =[

−Rd Rd −1]

, D1 =[

0 1]

, D2 = 0.

The transfer function from[

w u]T

to e is then

P =

[

A B1 B2

C1 D1 D2

]

.


Formulation as an H∞ problem

P

C u

e w

W

+

µ ξ v a

b P~

z~

y~

w~

Wd

ud

Break the loop involving the delay

Add the computational delay blockWdQ.-C. ZHONG: H


W =

[

Aw Bw

Cw 0

]

=

[

−ωc ωc

1 0

]

Wd =

[

Ad Bd

Cd Dd

]

=

[

− 2T

4T

1 −1

]

P =

A B2Cd 0 0 B1 B2Dd

0 Ad 0 0 0 Bd

BwC1 BwD2Cd Aw Bwξ BwD1 BwD2Dd

DwC1 DwD2Cd Cw Dwξ DwD1 DwD2Dd

0 0 0 0 0 µ

C1 D2Cd 0 ξ D1 D2Dd


System stabilityThe original closed-loop system is exponentially sta-ble if the designed closed-loop system is stable andits transfer function froma to b, denotedTba, satisfies‖Tba‖∞ < 1, where

Tba =

1 −

A B2

C1 D2

WdC

−1

W

=

A B2Cd B2DdCc 0 0

0 Ad BdCc 0 0

BcC1 BcD2Cd Ac + BcD2DdCc BcCw 0

0 0 0 Aw Bw

C1 D2Cd D2DdCc Cw 0

.


Design exampleParameters of the inverter

Parameter Value Parameter Value

Lf 150µH Rf 0.045Ω

Lg 450µH Rg 0.135Ω

Cf 22µF Rd 1Ω

42V DC voltage source

The generated three-phase voltage is connected tothe grid via a controlled circuit breaker and astep-up transformer.

The sampling frequency is5 kHz and the PWMswitching frequency is20 kHz.


Experimental setup


Circuit breaker

Measure 2 Measure 1 PCB

DC power source Inverter Output

filter

Transformer

dSpace 1104

da db dc i u

Local load

ug


Controller design

W =

[

−2550 2550

1 0

]

for f = 50Hz and

Wd =

[

−10000 20000

1 −1

]

.

ξ = 2.5 andµ = 0.8.

Using the MATLABhinfsyn algorithm, theH∞ con-troller C which nearly minimises theH∞ norm of thetransfer matrix fromw to z is obtained as

C(s) =735.2737(s + 1e004)(s2 + 9132s + 4.058e008)

(s + 1.109e004)(s + 2550)(s2 + 9515s + 4.232e008).

The resulting‖Tba‖∞ is 0.8426.Q.-C. ZHONG: H


Controller reduction

C(s) =735.2737(s + 1e004)(s2 + 9132s + 4.058e008)

(s + 1.109e004)(s + 2550)(s2 + 9515s + 4.232e008).

It can be reduced to

C(s) =735.27

s + 2550= KpW (s)

with Kp = 735.272550 without causing noticeable per-

formance degradation, after cancelling the poles andzeros which are close to each other. This leads to‖Tba‖∞ = 0.8222, which still maintains the stabilityof the system.


Resulting controller

uref plant

u

e ug

sde τ−

+

internal model

+ w

P

C

M

)(sW)(sW

pK

)(sW

uref plant

stabilizing

compensator

u

e ug

sdesW

τ−)(

+

internal model

+ w

p

P

C

M


Steady-state responses

-20

-10

0

10

20

Vo

ltag

e [V

]

0.00 0.01 0.02 0.03 0.04 0.05

Time [sec]

#1:1

#1:2

-4

-2

0

2

4

Vo

ltag

e er

ror

[V]

0.00 0.01 0.02 0.03 0.04 0.05

Time [sec]

#1:1

(a) voltageuA and its referenceuref (b) voltage tracking errore

-4

-2

0

2

4

Cu

rren

t [A

]

0.00 0.01 0.02 0.03 0.04 0.05

Time [sec]

#1:1

#1:2

(c) output currentiA and its reference

The recorded current THD was2.47%, while the output voltage

THD was1.74% and the grid voltage THD was2.11%.Q.-C. ZHONG: H


Transient responses

-4

-2

0

2

4

Cu

rren

t [A

]

3.60 3.65 3.70 3.75 3.80 3.85 3.90

Time [sec]

#1:1

#1:2

-20

-10

0

10

20

Vo

ltag

e [V

]

3.60 3.65 3.70 3.75 3.80 3.85 3.90

Time [sec]

#1:1

#1:2

(a) current outputiA and its referenceiref (b) voltage outputuA and its

referenceuref

-4

-2

0

2

4

Vo

ltag

e er

ror

[V]

3.60 3.65 3.70 3.75 3.80 3.85 3.90

Time [sec]

#1:1

(c) voltage tracking errore


H∞ repetitive current-controlled VSIsProposed control algorithms to improve total harmonic distortion usingH∞ and repetitive control.

Phase-lead low-pass

filter

DC power source

Inverter bridge

LC filter

Transformer

PWM modulation

Internal model M and stabilizing compensator C

Id* Iq*

iref e

abc

dq θ

Current controller

PLL

ugb uga ugc

u

+ +

+ +

+ +

u’gb u’ga

u’gc

u’

u’gb u’ga

u’gc

ia ib ic

- +

- +

- +

H∞ repetitive current-controlled VSIs

-3

-2

-1

0

1

2

3

Cu

rren

t [A

]

0.00 0.01 0.02 0.03 0.04 0.05

Time [sec]

#1:1

#1:2

The recorded current THD

was0.99%, while the grid

voltage THD was2.21%.


Synchronverters(Patent pending)

Synchronverters are inverters that are operated according

to the mathematical model of synchronous generators and

thus are grid-friendly.

Can work alone or in parallel without an external

communication channel.

Can work in grid-connected mode and island mode. In the

grid-connected mode, they can easily take part in the

regulation of real power and reactive power.

Time (Second)

P(W

)an

dQ

(Var

)

PXXy

Q


AC Ward Leonard drive systems (Patent pending)

Extended the concept of Ward Leonard drive systems to AC machines.

Constant speed

Variable speed

Controllable field Fixed field

Prime mover

Load

Variable speed

Variable speed

Fixed field

SM/IM Load

SG Prime mover VDC

Inverter

(a) Conventional (DC) Ward Leonard drive systems (b) AC WardLeonard drive systems

(c) Experimental results when reversing the motor: speed (left) and current (right)Q.-C. ZHONG: H


Parallel Operation of Inverters

Ro1 Ro2

1 1E δ∠ 2 2E δ∠1 1 1S P jQ= +0V∠

Z

2 2 2S P jQ= +

0 0.5 1 1.5 20

1000

2000

3000

P/ W

0 0.5 1 1.5 20

500

1000

1500

Q/ V

ar

t/s

P1

P2

Q2

Q1

(a) Joining of Inverter 2

0 0.5 1 1.5 20

1000

2000

3000

P/ W

0 0.5 1 1.5 20

500

1000

1500

Q/ V

ar

t/s

P2

P1

Q1

Q2

(b) Change of loadsQ.-C. ZHONG: H


Inverter-dominated Power SystemsGenerator 1

RenewableEnergy Source

EnergyStorage

Inverter

CircuitBreaker

Generator 2 Generator nRenewable

Energy Source

EnergyStorage

Inverter

CircuitBreaker

RenewableEnergy Source

EnergyStorage

Inverter

CircuitBreaker

AC BUS

Public Grid


A demonstration wind power system

Patented by Nheolis, France, installed on the department’srooftop

Control panelQ.-C. ZHONG: H


Buck Converter

Boost Converter


Regulation of induction generatorsfor wind power

Q

P


Energy recovery from landing aircraft

Coils

Risen slope to fall when energy recovery is activated

Aircraft

Runway Magnets with alternative poles (N, S, N, …)


Voltage and current (zoomed)

0 0.1 0.2 0.3 0.4 0.5-6000

-4000

-2000

0

2000

4000

6000

Pha

se A

vol

tage

0 0.1 0.2 0.3 0.4 0.5-1

-0.5

0

0.5

1x 10

5

Time

Pha

se A

cur

rent


0 5 10 15 20 25 30-6000

-4000

-2000

0

2000

4000

6000P

hase

A v

olta

ge

0 5 10 15 20 25 30-2000

-1000

0

1000

2000

Time

Pha

se A

cur

rent

(a) Phase current andthe generated voltage

(phase)

0200400600800

d

0

50

100

v

-10

-5

0

a

0

1

2x 10

7

p

0 5 10 15 20 25 300

5

10x 10

7

Time

E

(b) Distance, speed,deceleration, power and

energy


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