Download repetitive current controller for grid

H∞
REPETITIVE CURRENT CONTROLLER FOR
GRID - 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 support from the EPSRC, UK under the DTA scheme and
Q.-C. Zhong would like to thank the Royal Academy
of Engineering and the Leverhulme Trust for awarding
him a Senior Research Fellowship.
Dr Zhong would like to send his sincere apology for
having to cancel his trip at the last minute.
T. H ORNIK & Q.-C. Z HONG : 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. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 3/29
Motivation
Increasing share of renewable energy
UK: 20% by 2020
EU: 22% target for the share of renewable
energy sources and an 18% target for the
share of CHP in electricity generation by 2010
Regulation of system frequency and voltage
Threat to power system stability
Power quality
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 4/29
Power quality improvement
Power quality is an important problem for renewable
energy and distributed generation. The maximum total
harmonic distortion (THD) of output voltage allowed
is 5% (120V − 69kV ). The maximum THD allowed in
current 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. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 5/29
Current status
+
Ls , R s
ia
ea
VDC
va
vb
ib
eb
ic
ec
vc
Lg , R g
Circuit
Breaker
vga
vgb
vgc
C
-
Currently, most grid-connected inverters adopt the VSI topology with a current controller to regulate 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 multiple harmonic 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. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 6/29
Repetitive control
PI 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) :
ω
s2 +ω 2
Repetitive control: 1−e1−τd s , where τd is close to the signal
period. In order to guarantee the stability, a low-pass filter
1
W (s) is often added so the internal model is 1−W (s)e
−τd s .
e
+ +
p
e-τds
T. H ORNIK & Q.-C. Z HONG : H ∞
W(s)
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 7/29
Poles of the internal model
4
1
x 10
0.8
0.6
0.4
Im
0.2
0
*
true poles
o
approximated poles
−0.2
−0.4
−0.6
−0.8
−1
−18 −16 −14 −12 −10
T. H ORNIK & Q.-C. Z HONG : H ∞
−8
Re
−6
−4
−2
0
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 8/29
Objective of this talk
To design a current controller to minimise the
current THD, which is
equipped with the repetitive control technique
designed with the H ∞ control theory
To demonstrate the performance with
experimental results
Also to cover other issues, such as synchronisation
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 9/29
Overall structure of the system
Transformer
DC power
source
Inverter
bridge
LC
filter
ia
ib
uga ugb ugc
ic
u’ga
u’gb
PWM
modulation
u’gc
u’
+
u
Phase-lead
low-pass
filter
u’ga
+
+
PLL
u’gb
+
+
+
Internal model M
and stabilizing
compensator C
u’gc
e
+
-
θ
iref
+
-
+
dq
abc
Id*
Iq*
Current controller
Individual controllers are adopted for each phase in the natural abc 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 current iref , which is generated from the d, q-current
references Id∗ and Iq∗ using the dq → abc transformation.
A phase-locked loop (PLL) is used to provide the phase information of the grid voltage,
which is needed to generate iref .
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 10/29
Synchronisation
When the references Id∗ and Iq∗ are all equal to 0, 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 and ugc ) 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 than 1 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. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 11/29
H
∞
current repetitive control
P
w
ug
iref
W ( s )e
e +
plant
−τ d s
+
internal model
u
M
stabilizing
compensator
p
C
To minimise the tracking error e between the current
reference and the current injected to the grid.
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 12/29
Single phase representation
+ VDC PWM
u’
Inverter uf
bridge
Rf
Lf
i1
Cf
filter inductor
Sc
uo
ic
Lg
uc
i2
Rg
grid interface inductor
ug
grid
Rd
neutral
States: x =
i1 i2 uc
External signals: w =
T
ug iref
Controlled signal: e = iref − i2
T. H ORNIK & Q.-C. Z HONG : H ∞
T
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 13/29
State-space model
ẋ = Ax + B1 w + B2 u
y = e = C1 x + D 1 w + D 2 u
with


A=

R +R
− fLf d
Rd
Lg
1
Cf
C1 =
h
Rd
Lf
R +R
− gLg d
− C1f
0 −1 0
− L1f
1
Lg
0
i
,


0

,


1
B1 = 
−
 Lg
0
D1 =
h
0 1
i
0

1
Lf






0  , B2 =  0 
,
0
0
, D2 = 0.
The corresponding plant transfer function is then
−1
P = D1 D2 + C1 (sI − A)
B1 B2 .
T. H ORNIK & Q.-C. Z HONG : H ∞

REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 14/29
Internal model M
e
+ +
p
e-τds
W(s)
1
τd = τ − ,
ωc
where ωc is the cut-off frequency of the low-pass filter
ωc
and τ is the signal period.
W (s) = s+ω
c
In order to maintain the tracking performance of the
controller, a frequency adaptive mechanism could be
used (not presented here).
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 15/29
Formulation of the H
∞
problem
~
P
b
~z
ξ
~ v
w
a
W
µ
w
e
P
+
~y
u
C
To minimise the H ∞ norm of T z̃w̃ = F l(P̃ , C) from
w̃ = [ v w ]T to z̃ = [ z1 z2 ]T , after opening the
local positive feedback loop of the internal model and
introducing weighting parameters ξ and µ.
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 16/29
The closed-loop system can be represented as


z̃
ỹ


 = P̃ 
u = C ỹ,
w̃
u

,
The extended plant P̃ consists of the original plant P together with the lowpass filter W and weighting parameters ξ and µ. The additional parameters ξ
and µ are added to provide more freedom in design.

A

 B C
 w 1

P̃ = 
0



0

C1
0
0
B1
Aw
Bw ξ
Bw D 1
Cw
0
0
0
0
0
0
ξ
D1
B2


Bw D 2 


.
0



µ

D2
The stabilising controller C can be calculated using the well-known results on
H ∞ controller design for the extended plant P̃ .
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 17/29
Stability evaluation
Assume that the state-space realisation of the controller is
Ac Bc
C=
.
Cc D c
The closed-loop system is exponentially stable if the
closed-loop system designed above is stable and the
transfer function from a to b,




Tba = 


A + B 2 Dc C 1 B 2 C c B 2 Dc C w
B c C1
Ac
B c Cw
0
0
Aw
C1
0
Cw
satisfies kTba k∞ < 1.
T. H ORNIK & Q.-C. Z HONG : H ∞
0


0 

,
Bw 

0
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-phase inverters and has the capability to generate PWM voltages from a constant 42V DC voltage source. The generated three-phase voltage
is connected to the grid via a controlled circuit breaker and a step-up transformer. The grid voltage
and the current injected into the grid are measured for control purposes. The sampling frequency
of the controller is 5 kHz and the PWM switching frequency is 20 kHz.
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 19/29
Block diagram of the system
PCB
Measure 1
DC power
source
Inverter
bridge
da
db
Measure 2
Transformer
LC
filter
Circuit breaker
dc
i
ug
dSpace
1104
Inverter parameters
Parameter
Value
Parameter
Value
Lf
150µH
Rf
0.045Ω
Lg
450µH
Rg
0.135Ω
Cf
22µF
Rd
1Ω
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 20/29
H
∞
controller design
The low-pass filter W is chosen as, for f = 50Hz,
−2550 2550
.
W =
1
0
The weighting parameters are chosen to be ξ = 100
and µ = 0.25.
Using the MATLAB hinf syn algorithm, the H ∞ controller C which nearly minimises the H ∞ norm of the
transfer matrix from w̃ to z̃ is obtained as
15911809755.474(s + 300.8)(s2 + 9189s + 4.04 × 108 )
C(s) =
.
9
2
4
8
(s + 8.745 × 10 )(s + 2550)(s + 1.245 × 10 s + 3.998 × 10 )
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 21/29
Controller reduction
To replace s with 0 for very high-frequency
modes
To cancel the poles and zeros that are close to each
other.
The reduced controller is
1.8195(s + 300.8)
C(s) =
= W (s)CP D (s)
s + 2550
with
1.8195(s + 300.8)
CP D (s) =
.
2550
The resulting kTba k∞ is 0.4555 and, hence, the closedloop system is stable.
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 22/29
Phase (deg)
Magnitude (dB)
Comparison of the controllers
20
10
0
-10
-20
-30
-40
90
Original
Reduced
45
0
-45
-90
2
10
4
6
8
10
10
10
Frequency (rad/sec)
10
10
12
10
There is little difference at low frequencies. The Bode
plots in the discrete time domain are almost identical,
for the sampling frequency of 5kHz used for implementation.
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 23/29
The designed controller
e −τ d s
P
w
ug
iref
plant
e +
+
W (s )
internal model
M
u
CPD (s )
C
It is interesting to see that CP D (s) can actually be regarded as an inductor that converts the output (current)
signal from the internal model to a voltage signal u.
Using MATLAB c2d (ZOH) algorithm, the discretised
controller can be obtained as
1.8195(z − 0.9529)
.
C(z) =
z − 0.6005
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 24/29
Experimental results
Synchronisation process
Steady-state responses
Transient responses
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 25/29
Synchronisation process
As explained before, grid voltages (uga , ugb and ugc )
are feed-forwarded through a phase-lead low-pass filter and added to the control signal for the inverter to
synchronise with the grid. The inverter synchronisation process was started at around t = 2.837 second
and, immediately, it is synchronised and ready to be
connected to the grid.
Voltage [V]
10
#1:2
0
ug
20
Voltage error [A]
20
uA
?
#1:1
-10
-20
2.80
2.82
2.84
2.86
2.88
10
#1:1
0
-10
-20
2.80
2.84
2.86
2.88
Time [sec]
Time [sec]
(a) output voltage uA and grid voltage ug
T. H ORNIK & Q.-C. Z HONG : H ∞
2.82
(b) uA -ug
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 26/29
Steady-state responses
3
#1:1
1.0
#1:2
Current error [A]
Current [A]
2
iA
1
0
-1
Y iref
H
-2
-3
0.00
0.01
0.02
0.03
0.04
0.05
0.5
#1:1
0.0
-0.5
-1.0
Time [sec]
0.00
0.01
0.02
0.03
0.04
0.05
Time [sec]
(a) current output iA and its reference iref
(b) the current error e
The current reference Id∗ was set at 3A. This corresponds to 76.4W active power generated by the inverter. The reactive power was set at
0VAR (Iq∗ = 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 was 0.99%, while the grid voltage THD was
2.21%.
T. H
& Q.-C. Z
: H∞
– p. 27/29
ORNIK
HONG
REPETITIVE CURRENT CONTROLLER FOR GRID CONNECTED INVERTERS
Transient responses
iref
2
1.0
H
j
iA
Current error [A]
3
Current [A]
#1:1
1
0
#1:2
-1
0.5
0.0
#1:1
-0.5
-2
-3
4.05
4.10
4.15
4.20
4.25
-1.0
Time [sec]
4.05
4.10
4.15
4.20
4.25
Time [sec]
(a) current output iA and its reference iref
(b) the current error e
A step change in the current reference Id∗ from 2A to
3A was applied (while keeping Iq∗ = 0). The inverter
responded to the current step change in about 5 cycles.
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 28/29
Summary
The H ∞ 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 proposed H ∞ repetitive
current controller offers excellent performance with a recorded
current THD less than 1%.
T. H ORNIK & Q.-C. Z HONG : H ∞
REPETITIVE CURRENT CONTROLLER FOR GRID - CONNECTED INVERTERS
– p. 29/29