h repetitive current controller for grid...
Post on 25-Jul-2020
7 Views
Preview:
TRANSCRIPT
H∞ REPETITIVE CURRENT CONTROLLER FORGRID-CONNECTED INVERTERS
Tomas Hornik and Qing-Chang Zhong
Dept. of Electrical Eng. & Electronics
The University of Liverpool
UK
Email: Q.Zhong@liv.ac.uk
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
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 3/29
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
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 4/29
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%
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 5/29
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.
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 6/29
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
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 7/29
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
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 8/29
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
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 9/29
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 .
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 10/29
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.
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 11/29
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.
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 12/29
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
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 13/29
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
]
.
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 14/29
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).
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 15/29
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.
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 16/29
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 .
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 17/29
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
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 20/29
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).
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 21/29
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.
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 22/29
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.
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 23/29
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.
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 24/29
Experimental results
Synchronisation process
Steady-state responses
Transient responses
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 25/29
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.
T. HORNIK & Q.-C. ZHONG: H∞ REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS– p. 28/29
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
top related