h repetitive current controller for grid...

H∞ REPETITIVE CURRENT CONTROLLER FORGRID-CONNECTED INVERTERS

Tomas Hornik and Qing-Chang Zhong

Dept. of Electrical Eng. & Electronics

The University of Liverpool

UK

Email: [email protected]

Acknowledgement & apology

T. Hornik would like to acknowledge the financial sup-port from the EPSRC, UK under the DTA scheme andQ.-C. Zhong would like to thank the Royal Academyof Engineering and the Leverhulme Trust for awardinghim a Senior Research Fellowship.

Dr Zhong would like to send his sincere apology forhaving to cancel his trip at the last minute.

T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 2/29

Outline

Motivation

Brief introduction to repetitive control

Overall structure of the system

Synchronisation

H∞ controller design

Experimental setup and results

Summary


MotivationIncreasing share of renewable energy

UK: 20% by 2020EU: 22% target for the share of renewableenergy sources and an18% target for theshare of CHP in electricity generation by2010

Regulation of system frequency and voltage

Threat to power system stability

Power quality


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 shown below:

Odd harmonics Maximum current THD

< 11th < 4%

11th − 15th < 2%

17th − 21th < 1.5%

23rd − 33rd < 0.6%

> 33rd < 0.3%


Current status

+

-

Ls , Rs va

vb

vc

ia

ib

ic

ea

eb

ec

VDC

C

vga

vgb

vgc

Circuit Breaker

Lg , Rg

Currently, most grid-connected inverters adopt the VSI topology with a current controller to regu-

late the current injected into the grid by using schemes

Proportional-integral (PI) controllers in the synchronously rotating (d, q) reference frame:

works well with balanced systems, but cannot cope with unbalanced disturbance currents

Proportional-resonant (PR) controllers in the stationary(α, β) reference frame: popular

due to the capability of eliminating the steady state error,while regulating sinusoidal

signals, and the possible extension to compensate multipleharmonic but difficult to cope

with varying grid frequency.

Hysteresis controllers in the natural (abc) frame: simple and fast but it results in high and

variable sampling frequencies.


Repetitive controlPI controllers are good for tracking or rejecting step signals. But

for inverters, the signals are sinusoidal. In order to have good

tracking performance, a pair of conjugate poles on the imaginary

axis are needed.

Proportional-resonant (PR) :ωs2+ω2

Repetitive control: 11−e−τds , whereτd is close to the signal

period. In order to guarantee the stability, a low-pass filter

W (s) is often added so the internal model is 11−W (s)e−τds .

W(s) e-τds

+ + e p


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


Objective of this talk

To design a current controller to minimise thecurrent THD, which is

equipped with the repetitive control techniquedesigned with theH∞ control theory

To demonstrate the performance withexperimental results

Also to cover other issues, such as synchronisation


Overall structure of the system

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

- +

- +

- +

Individual controllers are adopted for each phase in the naturalabc frame.Equipped with a neutral point controller so that a balanced neutral point is available.It has a current loop including a repetitive controller so that the current injected into the

grid could track the reference currentiref , which is generated from thed, q-current

referencesI∗d

andI∗q using thedq → abc transformation.

A phase-locked loop (PLL) is used to provide the phase information of the grid voltage,

which is needed to generateiref .


SynchronisationWhen the referencesI∗

d andI∗

q are all equal to0, the generated voltage should

be equal to the grid voltage, i.e., the inverter should be synchronised with the

grid and the circuit breaker could be closed at any time if needed. In order to

achieve this, the grid voltages (uga, ugb andugc) are feed-forwarded and added

to the output of the repetitive current controller via a phase-lead low-pass filter

F (s) =33(0.05s + 1)

(s + 300)(0.002s + 1),

which has a gain slightly higher than1 and a phase lead at the fundamental

frequency. It is introduced to compensate the phase shift and gain attenuation

caused by computational delay, PWM modulation, the inverter bridge and the

LC filter. It also attenuates the harmonics in the feed-forwarded grid voltages.

simple (but effective)

improves the dynamics during grid voltage fluctuations

does not affect the independence of each phase.


H∞ current repetitive control

iref plant

stabilizing

compensator

u

e ug

sdesW

τ−)(

+

internal model

+ w

p

P

C

M

To minimise the tracking errore between the currentreference and the current injected to the grid.


Single phase representation

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

States:x =[

i1 i2 uc

]T

External signals:w =[

ug iref]T

Controlled signal:e = iref − i2


State-space modelx = Ax + B1w + B2u

y = e = C1x + D1w + D2u

with

A =

−Rf+Rd

Lf

Rd

Lf− 1

Lf

Rd

Lg−

Rg+Rd

Lg

1

Lg

1

Cf− 1

Cf0

, B1 =

0 0

− 1

Lg0

0 0

, B2 =

1

Lf

0

0

,

C1 =[

0 −1 0]

, D1 =[

0 1]

, D2 = 0.

The corresponding plant transfer function is then

P =[

D1 D2

]

+ C1(sI − A)−1[

B1 B2

]

.


Internal model M

W(s) e-τds

+ + e p

τd = τ −1

ωc

,

whereωc is the cut-off frequency of the low-pass filterW (s) = ωc

s+ωcandτ is the signal period.

In order to maintain the tracking performance of thecontroller, a frequency adaptive mechanism could beused (not presented here).


Formulation of the H∞ problem

P

C u

e w

W

+

µ ξ v a

b P~

z~

y~

w~

To minimise theH∞ norm of T zw = F l(P , C) fromw = [ v w ]T to z = [ z1 z2 ]T , after opening thelocal positive feedback loop of the internal model andintroducing weighting parametersξ andµ.


The closed-loop system can be represented as

z

y

= P

w

u

,

u = Cy,

The extended plantP consists of the original plantP together with the low-

pass filterW and weighting parametersξ andµ. The additional parametersξ

andµ are added to provide more freedom in design.

P =

A 0 0 B1 B2

BwC1 Aw Bwξ BwD1 BwD2

0 Cw 0 0 0

0 0 0 0 µ

C1 0 ξ D1 D2

.

The stabilising controllerC can be calculated using the well-known results on

H∞ controller design for the extended plantP .


Stability evaluationAssume that the state-space realisation of the con-troller is

C =

[

Ac Bc

Cc Dc

]

.

The closed-loop system is exponentially stable if theclosed-loop system designed above is stable and thetransfer function froma to b,

Tba =

A + B2DcC1 B2Cc B2DcCw 0

BcC1 Ac BcCw 0

0 0 Aw Bw

C1 0 Cw 0

,

satisfies‖Tba‖∞ < 1.T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 18/29

Experimental setup

It consists of an inverter board, a three-phase LC filter, a three-phase grid interface inductor, a

board consisting of voltage and current sensors, a step-up transformer, a dSPACE DS1104 R&D

controller board with ControlDesk software, and MATLAB Simulink/SimPower software package.

The inverter board consists of two independent three-phaseinverters and has the capability to gen-

erate PWM voltages from a constant42V DC voltage source. The generated three-phase voltage

is connected to the grid via a controlled circuit breaker anda step-up transformer. The grid voltage

and the current injected into the grid are measured for control purposes. The sampling frequency

of the controller is5 kHz and the PWM switching frequency is20 kHz.T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 19/29

Block diagram of the system

Circuit breaker

Measure 2 Measure 1 PCB

DC power source

Inverter bridge

LC filter

Transformer

dSpace 1104

da db dc i ug

Inverter parametersParameter Value Parameter Value

Lf 150µH Rf 0.045Ω

Lg 450µH Rg 0.135Ω

Cf 22µF Rd 1Ω


H∞ controller designThe low-pass filterW is chosen as, forf = 50Hz,

W =

[

−2550 2550

1 0

]

.

The weighting parameters are chosen to beξ = 100andµ = 0.25.

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

C(s) =15911809755.474(s + 300.8)(s2 + 9189s + 4.04 × 108)

(s + 8.745 × 109)(s + 2550)(s2 + 1.245 × 104s + 3.998 × 108).


Controller reductionTo replaces with 0 for very high-frequencymodes

To cancel the poles and zeros that are close to eachother.

The reduced controller is

C(s) =1.8195(s + 300.8)

s + 2550= W (s)CPD(s)

with

CPD(s) =1.8195(s + 300.8)

2550.

The resulting‖Tba‖∞ is 0.4555 and, hence, the closed-loop system is stable.


Comparison of the controllers

-40-30-20-10

01020

Mag

nitu

de (

dB)

102

104

106

108

1010

1012

-90

-45

0

45

90P

hase

(de

g)

Frequency (rad/sec)

OriginalReduced

There is little difference at low frequencies. The Bodeplots in the discrete time domain are almost identical,for the sampling frequency of5kHz used for imple-mentation.


The designed controller

iref plant

u

e ug

sde τ−

+

internal model

+ w

P

C

M

)(sW

)(sCPD

It is interesting to see thatCPD(s) can actually be re-garded as an inductor that converts the output (current)signal from the internal model to a voltage signalu.Using MATLAB c2d (ZOH) algorithm, the discretisedcontroller can be obtained as

C(z) =1.8195(z − 0.9529)

z − 0.6005.


Experimental results

Synchronisation process

Steady-state responses

Transient responses


Synchronisation processAs explained before, grid voltages (uga, ugb andugc)are feed-forwarded through a phase-lead low-pass fil-ter and added to the control signal for the inverter tosynchronise with the grid. The inverter synchronisa-tion process was started at aroundt = 2.837 secondand, immediately, it is synchronised and ready to beconnected to the grid.

-20

-10

0

10

20

Vo

ltag

e [V

]

2.80 2.82 2.84 2.86 2.88

Time [sec]

#1:1

#1:2uA

?

ug

-20

-10

0

10

20

Vo

ltag

e er

ror

[A]

2.80 2.82 2.84 2.86 2.88

Time [sec]

#1:1

(a) output voltageuA and grid voltageug (b) uA-ugT. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 26/29

Steady-state responses

-3

-2

-1

0

1

2

3C

urr

ent

[A]

0.00 0.01 0.02 0.03 0.04 0.05

Time [sec]

#1:1

#1:2

irefHY

iA

-1.0

-0.5

0.0

0.5

1.0

Cu

rren

t er

ror

[A]

0.00 0.01 0.02 0.03 0.04 0.05

Time [sec]

#1:1

(a) current outputiA and its referenceiref (b) the current errore

The current referenceI∗d was set at 3A. This corresponds to76.4W ac-

tive power generated by the inverter. The reactive power wasset at

0VAR (I∗q = 0). This corresponds to the unity power factor. Since there

is no local load included in the experiment, all generated active power

was injected into the grid via a step-up transformer.

The recorded current THD was0.99%, while the grid voltage THD was

2.21%.T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 27/29

Transient responses

-3

-2

-1

0

1

2

3C

urr

ent

[A]

4.05 4.10 4.15 4.20 4.25

Time [sec]

#1:1

#1:2

irefHj iA

-1.0

-0.5

0.0

0.5

1.0

Cu

rren

t er

ror

[A]

4.05 4.10 4.15 4.20 4.25

Time [sec]

#1:1

(a) current outputiA and its referenceiref (b) the current errore

A step change in the current referenceI∗d from 2A to3A was applied (while keepingI∗q = 0). The inverterresponded to the current step change in about 5 cycles.


SummaryTheH∞ repetitive control strategy has been applied to the design

of a current controller for grid-connected inverters. The resulting

controller is simple and consists of an internal model and a

proportional-derivative controller.

It has shown that advanced control theories can be applied to

design implementable controllers for practical applications and

can offer insightful understanding to real problems.

A simple and effective synchronisation mechanism has also been

introduced for the proposed control strategy to quickly

synchronise the inverter with the grid.

Experimental results have shown that the proposedH∞ repetitive

current controller offers excellent performance with a recorded

current THD less than1%.T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 29/29

h repetitive current controller for grid...

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