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Modulation and current/voltage control of the grid converter
Modulation and current/voltage control
of the grid converter
Marco Liserre
[email protected]
Marco Liserre
[email protected]
Modulation and current/voltage control of the grid converter
Introduction

Modulation and ac current control are the core of grid-connected converters

They are responsible of the safe operation of the converter and of the compliance
with standards and grid codes

Ac voltage control is a standard solution in WT-system however can be adopted
also in PV-system for reinforcing stability or offering ancillary services
A glance at the lecture content





Introduction
Model of the grid converter
Overview of modulation techniques
Current control
Voltage control
Marco Liserre
[email protected]
Modulation and current/voltage control of the grid converter
Introduction: modulation and current/voltage control
i
VSI
C
L1
L2
+
+
vdc
v
-
Modulator
Cf
v*
vg ig Lg
+
+
-
-
vC
e
Current
control

PI-based current control implemented in a synchronous frame is commonly used in
three-phase converters

In single-phase converters the PI controller capability to track a sinusoidal
reference is limited and Proportional Resonant (PR) can offer better performances

Modulation has an influence on design of the converter (dc voltage value), losses
and EMC problems including leakage current
Marco Liserre
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Modulation and current/voltage control of the grid converter
Introduction: harmonic limits for PV inverters
 In Europe there is the standard IEC 61727
 In US there is the recommendation IEEE 929
 the recommendation IEEE 1547 is valid for all distributed resources technologies
with aggregate capacity of 10 MVA or less at the point of common coupling
interconnected with electrical power systems at typical primary and/or secondary
distribution voltages
 All of them impose the following conditions regarding grid current harmonic content
The total THD of the grid current should not be higher than 5%
Marco Liserre
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Modulation and current/voltage control of the grid converter
Introduction: harmonic limits for WT inverters
In Europe the standard 61400-21 recommends to apply the standard 61000-3-6 valid for
polluting loads requiring the current THD smaller than 6-8 % depending on the type of
network.
harmonic
5th
7th
11th
13th
limit
5-6 %
3-4 %
1.5-3 %
1-2.5 %
in case of several WT systems
 I hi 

  
i 1  i 
N
Ih 

in WT systems asynchronous and synchronous generators directly connected to
the grid have no limitations respect to current harmonics
Marco Liserre
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Modulation and current/voltage control of the grid converter
Model of the grid converter
id
TA
DA
TB 
DB 
TC 
DC 
A
vdc
+
-
io
B
R
L
e
n
C
TA
DA
TB 
DB 
TC 
DC 
converter switching function
p(t)  2  pa(t)    pb(t)   2  pc(t) 
3
ac voltage equation
d i (t ) 1
 Ri (t )  e (t )  p(t )vdc (t ) 
dt
L
Marco Liserre
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Modulation and current/voltage control of the grid converter
Use of a synchronous frame
c

 p  2 1 1 2
p   
   3 0  3 2
q
v
v
vq
v
vd
b
 di  t 


 dt

 di  t  
 dt
Marco Liserre
d


a

cos 
 pd  
p   
 q   sin 


-frame
1
  Ri  t   e  t   p  t  vdc  t  
L
1
  Ri  t   e  t   p  t  vdc  t  
L
p 
1 2   a 
  pb 
3 2
 pc 
2 
2  


cos   
cos




  pa 
3 
3    


pb 


2 
2 


sin   
 sin   
   pc 
3
3




dq-frame
 did  t 
1
 iq  t     Rid  t   ed  t   pd  t  vdc  t  

 dt
L

 diq  t   i t  1   Ri t  e t  p t v t 
d  
q 
q 
q   dc   
 dt
L
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Modulation and current/voltage control of the grid converter
Overview of modulation techniques
Marco Liserre
[email protected]
Modulation and current/voltage control of the grid converter
Modulation techniques
 Characteristic parameters of these strategies are:
 the ratio between amplitudes of modulating and carrier waves (called
modulation index M)
 the ratio between frequencies of the same signals (called carrier index m)
 These techniques differ for the modulating wave chosen with the goal to obtain
 a lower harmonic distortion,
 to shape the harmonic spectrum
 to guarantee a linear relation between fundamental output voltage and
modulation index in a wider range
 The space vector modulations are developed on the basis of the space vector
representation of the converter ac side voltage
Marco Liserre
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Modulation and current/voltage control of the grid converter
Modulation techniques
 analogic or digital,
 natural sampled or regular sampled
 symmetric or asymmetric
VAo 1
 Vd 
 
 2 
4
  1.278 
1.0
Linear
modulation
0
Marco Liserre
Squarewave
Over-
Optimization both for the
linearity and harmonic content
0 1.0
 for m
3.24
f
 15

M
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Modulation and current/voltage control of the grid converter
Sinusoidal PWM (SPWM)
Vˆcontrol
Amplitude modulation ratio : M 
Vˆtri
f
Frequency modulation ratio : m  S
f1
+
Vdc
V
-
Vdc +
C1 P
2 -
-
Output voltage averaged over one
switching period:
VAo  vAo 
Marco Liserre
vcontrol Vd
; vcontrol  Vˆtri
V tri 2
+
P
Vdc +
C2
2 + v t -
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Modulation and current/voltage control of the grid converter
Sinusoidal PWM (SPWM)
Assuming a sinusoidal control signal:
vcontrol  Vˆcontrol sin(1t ),
the fundamental frequency component
of the output voltage is given by:
 vAo 1  M sin(1t )
 
VˆAo
1
M
Vd
2
Vd
2
M  1.0
M  1.0
The inverter stays in its “linear range”
while M   0,1.
The harmonics in the output voltage appear as
sidebands of fS and its multiples
f h   jm  k  f1
1
hth
The
harmonic corresponds to the
of j times the frequency modulation ratio m.
For even values of j only exist harmonics for
odd values of k, and viceversa.
Marco Liserre
15
kth sideband
13 17
29 31
27
33
25
35
43454749
3941
51
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Modulation and current/voltage control of the grid converter
Bipolar and unipolar modulations
+
Vdc
P
Vdc +
C2
2 + v t -
-
Vdc +
C1 P
2 -
 bipolar
 unipolar
Marco Liserre
v(t ) 
4Vdc




m0 n1
m0 n
+
P
P
+ v t -
Vdc
P
P
-
1    

J n  q M  sin m  n  cosmc t  n0t 
q  2  
2
v(t )  2Vdc M cos0t  
8Vdc



  2m J
1
2n1
mM cosm  n 1 cos2mct  2n 10t 
m1 n
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Modulation and current/voltage control of the grid converter
Bipolar and unipolar modulations
1.5
1
1.5
1
vref , vtri
vref , vtri
0.5
0
-0.5
-1
vref
-1.5
0
10
20
t [ms]
100
vtri
30
0.5
0
-0.5
-1
40
vref
-1.5
0
100
vtri
10
20
t [ms]
30
40
10
20
t [ms]
30
40
10
20
t [ms]
30
40
10
20
t [ms]
30
40
50
vAo [V]
vAo [V]
50
0
-50
0
-50
-100
0
100
10
20
t [ms]
30
40
-100
0
100
50
vBo [V]
50
vBo [V]
0
-50
0
-50
-100
0
150
100
10
20
t [ms]
30
40
vAB [V]
vAB [V]
50
0
-50
-100
0
-50
10
20
t [ms]
30
40
-150
0
vAB/Vd
1
0.8
0.6
0.4
0.2
0
0
50
-100
-150
0
vAo/Vd
-100
0
150
100
5
10
15
20
25
30
h
35
40
45
50
55
60
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
h
35
40
45
50
55
60
Due to the unipolar PWM the odd carrier and associated sideband harmonics are
completely cancelled leaving only odd sideband harmonics (2n-1) terms and even (2m)
carrier groups
Marco Liserre
[email protected]
Modulation and current/voltage control of the grid converter
Three-phase modulation techniques
1.5
1
100
50
vAn [V]
vref , vtri
0.5
0
-0.5
-1
-1.5
0
100
vref
10
20
t [ms]
30
-50
vtri
-100
0
150
100
40
0
10
20
t [ms]
30
40
-150
0
20
10
20
t [ms]
30
40
10
20
t [ms]
30
40
10
20
t [ms]
30
40
10
io [A]
von [V]
40
-100
0
-50
0
-10
10
20
t [ms]
30
-20
0
20
40
10
id [A]
50
vAB [V]
30
0
50
0
-50
0
-10
-100
-150
0
20
t [ms]
-50
-50
-100
0
150
100
10
50
E [V]
vAo [V]
50
-100
0
100
0
10
20
t [ms]
30
40
-20
0
The basic three-phase modulation is obtained applying a bipolar modulation to each of
the three legs of the converter. The modulating signals have to be 120 deg displaced.
The phase-to-phase voltages are three levels PWM signals that do not contain triple
harmonics. If the carrier frequency is chosen as multiple of three, the harmonics at the
carrier frequency and at its multiples are absent.
Marco Liserre
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Modulation and current/voltage control of the grid converter
Extending the linear range (m=1,1)
SPWM
SVM
Marco Liserre
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64
Modulation and current/voltage control of the grid converter
Three-phase continuous
modulation techniques
Continuous modulations
 sinusoidal PWM with Third Harmonic Injected –THIPWM. If the third
harmonic has amplitude 25 % of the fundamental the minimum current
harmonic content is achieved; if the third harmonic is 17 % of the fundamental
the maximal linear range is obtained;
 suboptimum modulation (subopt). A triangular signal is added to the
modulating signal. In case the amplitude of the triangular signal is 25 % of the
fundamental the modulation corresponds to the Space Vector Modulation
(SVPWM) with symmetrical placement of zero vectors in sampling time.
v ACR
v ACR
THIPWM1/6
vdc /2
vDCR
0
vDCR
0
0
von
von
-vdc /2
von
-vdc /2
-vdc /2
0
/2

2/3
Triple harmonic injection 1/6
Maximum linear range
2
SVPWM
vdc /2
vdc /2
vDCR
Marco Liserre
v ACR
THIPWM1/4
0
/2

2/3
Triple harmonic injection 1/4
Minimum current harmonics
2
0
/2

2/3
2
Space vector - Triangular 1/4
Maximum linear range and almost optimal
current harmonics
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Modulation and current/voltage control of the grid converter
Three-phase discontinuous
modulation techniques
The discontinuous modulations formed by unmodulated 60 deg segments in order to
decrease the switching losses
symmetrical flat top modulation, also called DPWM1;
asymmetrical shifted right flat top modulation, also called DPWM2;
asymmetrical shifted left flat top modulation, also called DPWM0.
v ACR
vdc /2
vDCR
DPWM0
=-/6
vDCR
vdc /2
0
von
-vdc /2
/2
v ACR
von
-vdc /2

2/3
2
DPWM2
=/6
0
von
0
vDCR
vdc /2
v ACR
0
Marco Liserre
DPWM1
=0
-vdc /2
0
/2

2/3
2
0
/2

2/3
2
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Modulation and current/voltage control of the grid converter
Multilevel converters and modulation techniques
 Wind turbine systems: high
power -> 5 MW Alstom
converter
 Photovoltaic systems: many dclinks for a transformerless
solution

Vdc
2

0V
0V
0V

Vdc
2

Different possibilities:
A
C
B
 alternative phase opposition (APOD) where carriers in adjacent bands
are phase shifted by 180 deg;
 phase opposition disposition (POD), where the carriers above the
reference zero point are out of phase with those below zero by 180 deg;
 phase disposition (PD), where all the carriers are in phase across all
bands.
Marco Liserre
[email protected]
Modulation and current/voltage control of the grid converter
Multilevel converters and modulation techniques
Neutral point clamped
Cascaded inverter


i
a
Vdc
b
L
Vdc




n

e
v


c
Vdc

v
Vdc
d
L
a i

e

b



v(t )  NVdc M cos0t  


N


1

J 2n1 mM cosm  n  1  cos2mc t  2n  10t  2m i 
 m1 n 2m
i 1
4Vdc
• carrier shifting
Marco Liserre
i 
(i  1)
N
m  kN , k  1, 2,3...
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Modulation and current/voltage control of the grid converter
v ,V
tri
Carrier shifting
1
ref
0
-1
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
0.06
t [s]
0
v
ab
[V]
100
-100
ref
v ,V
tri
0.02
0.03
0.035
0.04
t [s]
0.045
0.05
0.055
0.06
0.025
0.03
0.035
0.04
t [s]
0.045
0.05
0.055
0.06
0.025
0.03
0.035
0.04
t [s]
0.045
0.05
0.055
0.06
0.025
0.03
0.035
0.04
t [s]
0.045
0.05
0.055
0.06
1
0
-1
0.02
100
0
v
cd
[V]
0.025
-100
200
0
v
ad
[V]
0.02
-200
A [pu]
0.02
1
0.5
0
Marco Liserre
0
10
20
30
h
40
50
60
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Modulation and current/voltage control of the grid converter
PD Modulation for NPC
tri
1
ref
, V
0
V
ref
, V
tri
V
an
[V]
V
-1
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
0.06
1
0
-1
0.02
0.025
0.03
0.035
0.04
t [s]
0.045
0.05
0.055
0.06
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
0.06
1
0
-1
0.02
0.025
0.03
0.035
0.04
t [s]
0.045
0.05
0.055
0.06
200
0
-200
0.02
0.025
0.03
0.035
0.04
t [s]
0.045
0.05
0.055
0.06
1
0
A [pu]
V
ab
[V]
V
bn
[V]
-1
0.5
0
Marco Liserre
Best WTHD !
1
0
10
20
30
h
40
50
60
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Modulation and current/voltage control of the grid converter
Current Control
PWM current control methods
ON/OFF controllers
Separated PWM
linear
non-linear
passivity
PI
hysteresis
Marco Liserre
Delta
optimized
predictive
feedforward
fuzzy
resonant
dead-beat
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Modulation and current/voltage control of the grid converter
PI current control
 Typically PI controllers are used for the current loop in grid inverters
 Technical optimum design (damping 0.707 overshoot 5%)
--
+
v
-
Gd ( s )
+
GPI ( s )
+
i
+
e
Gf ( s )
GPI ( s )  K P 
ig
Gd ( s ) 
1
1  1.5Ts s
G f ( s) 
i( s)
1

v( s) R  Ls
vg
0
Magnitude (Db)
1
0.8
0.6
0.4
0.2
-5
-10
-15
-20 -1
10
0
10
0
10
1
10
2
10
3
10
4
0
Phase (Degree)
-0.2
-0.4
-0.6
-0.8
-1
-100
-200
-300
-400
-1
Marco Liserre
KI
s
-0.5
0
0.5
1
10
0
1
2
10
10
Frequency (Hz)
10
3
10
4
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Modulation and current/voltage control of the grid converter
Shortcomings of PI controller
1
reference
0.8
steady-state
magnitude and
phase error
0.98
0.96
0.94
0.92
0.9
0.88
0.86
0.84
0.82
0.8
0.023 0.0235 0.024 0.0245 0.025 0.0255 0.026 0.0265 0.027 0.0275
0.5
0
0.6
0.2
0.15
0.1
limited
disturbance
rejection
capability
0.4
0.2
0.05
0
-0.05
-0.1
actual
-0.5
error (scaled)
-0.15
-0.2
-0.25
0.019 0.0192 0.0194 0.0196 0.0198 0.02 0.0202 0.0204 0.0206 0.0208 0.021
reference
-1
error
0
-0.2
0
1
1
actual
-1.5
0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
time [s]
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
time [s]
 When the current controlled inverter is connected to the grid, the phase error results in
a power factor decrement and the limited disturbance rejection capability leads to the
need of grid feed-forward compensation.
 However the imperfect compensation action of the feed-forward control due to the
background distortion results in high harmonic distortion of the current and
consequently non-compliance with international power quality standards.
Marco Liserre
[email protected]
Modulation and current/voltage control of the grid converter
Use of a PI controller in a rotating frame


q
d
i (t )
e(t )
iq
The voltage used for
the dq-frame
orientation could be
measured after a
dominant reactance

e' t   e t   Lg
Marco Liserre

dig
dt

The current control can
be performed on the
grid current or on the
converter current
e‘(t)
e(t)
e‘(t)
id
DPI (s)dq
Ki

K

 p s

 0




Ki 
Kp  
s 
0
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Modulation and current/voltage control of the grid converter
Use of a PI controller in a rotating frame
id
id
i
αβ
i
id
e
i
 j
-+
Vg
αβ v g

v abc
e
iq
Kp 
Ki
s

v
v*
αβ


Current
controller
j
v gq

PLL
v g
Ki
s
L
abc
vg
Kp 
v gd
L
iq

f
-+
Current
controller
abc
v gd
e  j v gq
vdc
• active and reactive
Vdc controller

dc
V
K
Kp  i
s
+
-
P
power control can be
P controller
v
Q
Marco Liserre
P
P  v ga ia  v gb ib  v gc ic
Q
1
3
v
i  v gbc ib  v gca ic
gab a
PQ controller

Q
-+
-+
Q controller
-
+
i
K
Kp  i
s
K
Kp  i
s
id
iq
achieved
• vdc control can be
achieved too
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Modulation and current/voltage control of the grid converter
Use of a PI controllers in a rotating frame in
single-phase systems
id
id
 an independent Q control is
achieved
i
i
 A phase delay block create the
virtual quadrature component that
allows to emulate a two-phase
system
e  j
2
i
id
e
 j
e  j
2
-+
Vg
iq

PLL
v g
Ki
s

v
v
e j v 
L
v g
vg
Kp 
v gd
L
iq

f
-+
Current
controller
Kp 
Ki
s


Current
controller
v gq
v gd
e  j v gq
vdc
 the
v
component
of
the
idc
command voltage is ignored for
the calculation of the duty-cycle
x
Q

P
Vdc controller
K
Kp  i
s
-
+
MPPT
Vdc

v gd
Q


id
iq
v gq
Marco Liserre
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Modulation and current/voltage control of the grid converter
Use of a PI controllers in two rotating frames

 Under unbalanced conditions in order
compensate
the
harmonics
i
+
+
to
positive-
and
negative-sequence
+
iq

reference frames are required
e j
+
i

this
approach,
i
double computational effort must be
devoted
i
+

Marco Liserre

v
PI
e  j
e j
+
using
v
PI
id
+
 Obviously
v
PI
e  j
generated by the inverse sequence
present in the grid voltage both the
id
+
iq
PI
v

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Modulation and current/voltage control of the grid converter
Dead-beat controller
 The dead-beat controller belongs to the family of the predictive controllers
 They are based on a common principle: to foresee the evolution of the controlled
quantity (the current) and on the basis of this prediction:
 to choose the state of the converter (ON-OFF predictive) or
 the average voltage produced by the converter (predictive with pulse width
modulator)
 The starting point is to calculate its derivative to predict the effect of the control
action
 The controller is developed on the basis of the model of the filter and of the grid,
which is used to predict the system dynamic behavior: the controller is inherently
sensitive to model and parameter mismatches
Marco Liserre
[email protected]
Modulation and current/voltage control of the grid converter
Dead-beat controller
 The information on the model is used to decide the switching state of the
converter with the aim to minimize the possible commutations (ON-OFF
predictive) or the average voltage that the converter has to produce in order to null
e
 In case it is imposed that the error at the
GDB ( z )
+
i
--
end of the next sampling period is zero
v
-
+
it.
Gf ( z )
i
vg
the controller is defined as “dead-beat”. It
can be demonstrated that it is the fastest
current controller allowing nulling the
error after two sampling periods.
i*
i
tON
tON
k
Marco Liserre
Ts
k 1
Ts
k2
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Modulation and current/voltage control of the grid converter
Dead-beat controller
1
a
v(k  1)  v(k 1)  i(k )  i(k 1)  e(k  1)  e(k 1)
b
b
1
v(k  1)  vk   i(k )  e(k  1)  e(k )
b
reference
1
actual
response
response
0.8
0.6
0.4
0.4
0.2
0
0
470
480
490
500
510
samples
Marco Liserre
520
530
540
550
actual
0.6
0.2
460
reference
1
0.8
-0.2
450
neglecting R !
-0.2
450
460
470
480
490
500
510
520
530
540
550
samples
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Modulation and current/voltage control of the grid converter
20
20
10
10
0
0
-10
-10
current [A]
current [A]
Dead-beat controller: limits
-20
-30
-20
-30
-40
-40
-50
-50
-60
0.04
0.045
0.05
time [s]
0.055
-60
0.04
0.06
due to PWM !
0.045
0.05
time [s]
0.055
0.06
Pole-Zero Map
20
1
0.6/T
0.1
0.3/T
0.2
0.3
0.4
0.2/T
0.5
0.6
0.7
0.1/T
0.8
0.9
0.8/T
0.4

0.9/T
0.2
0
10
0.4/T
0.7/T
0.6
Imaginary Axis
0.5/T
0
current [A]
0.8
/T
/T
-20
-30
-0.2
0.9/T
0.1/T
-40
-0.4
0.8/T
-0.6
0.2/T
0.7/T
-0.8
-1
-1
-0.8
-0.6
-0.4
-50
0.3/T
0.6/T
Marco Liserre
-10
-0.2
0.5/T
0
Real Axis
-60
0.04
0.4/T
0.2
0.4
0.6
0.8
due to
parameter
error !
0.045
0.05
time [s]
0.055
0.06
1
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Modulation and current/voltage control of the grid converter
Resonant control
 Resonant control is based on the use of Generalized Integrator (GI)
 A double integrator achieves infinite gain at a certain frequency, called
resonance frequency, and almost no attenuation outside this frequency
GI
s
s2   2
 The GI will lead to zero stationary error and improved and selective
disturbance rejection as compared with PI controller
Bode Diagram
GI res pons e
2
s
s2   2
100
0
-100
-200
180
Phase (deg)
90
0
-90
-180
1
10
2
10
Frequency (Hz)
Marco Liserre
3
10
input and output of the resonant controller
Magnitude (dB)
200
in
out
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
time [s ec]
time
[s]
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Modulation and current/voltage control of the grid converter
Resonant control
 The resonant controller can
obtained via a frequency shift
GDC (s) 
Ki
s
GAC (s) 
GDC ( s ) 
Ki
1  ( s c ) 
G AC ( s) 
2K i s
s2  2
2 K ic s
s 2  2c s   2
Phase (deg)
Magnitude (dB)
80
600
400
200
2
10
Frequency (Hz)
60
40
20
0
1
10
3
10
100
100
50
50
Phase (deg)
Magnitude (dB)
800
0
1
10
G AC ( s)  GDC ( s  j )  GDC ( s  j )
be
0
-50
-100
1
10
2
10
Frequency (Hz)
3
10
2
3
10
Frequency (Hz)
10
0
-50
-100
1
10
2
3
10
Frequency (Hz)
10
Bode plots of ideal and non-ideal PR with KP = 1, Ki = 20,  = 314 rad/s c = 10 rad/s
Marco Liserre
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Modulation and current/voltage control of the grid converter
Resonant control
 The stability of the system should be taken into consideration
 The phase margin (PM) decreases as the resonant frequency approach to the
crossover frequency
s  1 

k

k
 P I 2


s   2  R  Ls 

s
kP  kI 2
s  2
Bode Diagram
Bode Diagram
400
400
Magnitude (dB)
300
200
100
0
200
100
-100
0
-200
180
-100
180
90
90
Phase (deg)
Phase (deg)
Magnitude (dB)
300
0
-90
-180
1
10
-90
2
10
Frequency (Hz)
Marco Liserre
0
3
10
-180
1
10
PM
2
10
3
10
Frequency (Hz)
[email protected]
Modulation and current/voltage control of the grid converter
Tuning of resonant control
 The gain Kp is founded by ensuring the desired bandwidth
 The integral constant Ki acts to eliminate the steady-state phase error
50Hz response
50Hz response
s
Gc ( s )  K P  K I 2
s  o2
6
4
4
2
2
Amp litude
Amp litude
6
0
-2
-4
-4
0
0.005
0.01
0.015
0.02
0.025
Time (sec)
Ki = 100
0.03
0.035
0.04
0. 045
0.05
s
s  o2
2
0
-2
-6
Gc ( s )  K P  K I
-6
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0. 045
0.05
Time (sec)
Ki = 500
 A higher Ki will "catch" the reference faster but with higher overshoot
 Another aspect is that Ki determines the bandwidth centered at the resonance
frequency, in this case the grid frequency, where the attenuation is positive. Usually, the
grid frequency is stiff and is only allowed to vary in a narrow range, typically ± 1%.
Marco Liserre
[email protected]
Modulation and current/voltage control of the grid converter
Discretization of generalized integrators
GI integrator decomposed in two simple
integrators
y s
s
 2
u s s  2
1

 y  s   s u  s   v  s  
 
v  s   1   2  y  s 

s
Forward integrator for direct path and backward for
feedback path
 yk  yk 1  Ts  (uk 1  vk 1 )

2
vk  vk 1  Ts    yk
y
u

v

2
The inverter voltage reference
K s 

ui* ( s )    s    K p  2 I 2 
s  

Control diagram of PR implementation
kp
Difference equations
 yk  yk 1  Ts  K I   k 1  Ts  vk 1
 *
ui ,k  K p   k  yk
v  v  T   2  y
k 1
s
k
 k
 k 1   k
y  y
k
 k 1
vk 1  vk
Marco Liserre

ii*
aw
u
y

kI
ii
ui*
Gf(s)
ii
v

2
, y  ymax
 ymax  y
aw  
 ymax  y , y   ymax
[email protected]
Modulation and current/voltage control of the grid converter
Use of P+resonant controller in stationary frame
idd*
ej

ixd*
Kp
+
-
+
+
2
i
3
The voltage used for reference
generation could be measured
after a dominant reactance
Ki  s
s2   2
ix*
ix
+
ux*
uy*
iy

SVM
6
Kp
+
+
u
PLL

Ki  s
s2   2
iy*
2
i*
3
DPR (s)
Marco Liserre
+
i (t )
e(t )
ix*

iy*
Ki s


K

0
p


s 2  02




Ki s 
0
K



p
s 2  02 

The current control can be
performed on the grid current
or on the converter current
[email protected]
Modulation and current/voltage control of the grid converter
PI vs PR for single-phase grid inverter current control
The current loop of PV inverter with PI
controller
GPI ( s )
v
-
Gd ( s )
+
+
--
+
+
i

Gf ( s )
i
ig
--
GP  RES ( s )
v
Gd ( s )
-
+
e

+
The current loop of PV inverter with
PR controller
e
i
Gf ( s )
vg
PI
G PI ( s )  K P 
KI
s
Gc (s)  K P  K I
PR
s
s   o2
2
1
Inverter Gd ( s )  sT  1
s
.
Plant


2
2
ii ( s)
1 s  zLC
G f ( s) 

2
ui ( s ) L fi s s 2  res


2
z LC

1
Lg C f

2
res
L

g
 Li   z LC
2
Li
 No grid voltage feed-forward is required
 GIs tuned to the low harmonics can be used for selective harmonic compensation
by cascading the fundamental component GI
Marco Liserre
[email protected]
Modulation and current/voltage control of the grid converter
From PI in a rotating-frame to P+res for each phase
In the hypothesis
Ed(s)
Vd(s)
+
 H11(s)=H22(s)= K p 
H11(s)
+
H12(s)
 H12(s)=H21(s)=0
vd (t )  h11 (t )  ed (t )

vq (t )  h22 (t )  eq (t )
H21(s)
+
Eq(s)
H22(s)
Gc
Marco Liserre
( d ,q )
Ki
s
+
Ki

K

 p s

 0

Vq(s)


Ki 
Kp  
s 
0


K p K s  3K 
K p K s  3K  
Ki s
  i 2 i2 0   i 2 i2 0 
 Kp  2
2
2 2  (s   0 )
2 2  (s   0 ) 
s  0

K p K s  3K  
Ks
2  K p K s  3K 
( a , b .c )
Gc ( s)     i 2 i 2 0
Kp  2 i 2
  i 2 i2 0 
3  2 2  (s   0 )
2 2  (s   0 ) 
s  0
K p K s  3K 
 K p K i s  3K i 0

Ks
  i 2 i2 0
Kp  2 i 2 
 
2
2
2 2  (s   0 )
s  0
 2 2  (s   0 )

[email protected]
Modulation and current/voltage control of the grid converter
Linear controllers : from PI in a rotating-frame to P+res for each phase
va
d ia (t ) 1  R

dt ic (t )  L  0
z
va (t )
0  ia (t ) 1 2  1  1 

 
 vb (t ) 



R  ic (t )  3 1 1
2
 vc (t ) 
va (t )  vb (t )  vc (t )  0
vc
vb
z
z
d ia (t ) 1  R

dt ic (t )  L  0
0  ia (t ) 1 0  va (t )



R ic (t )  0  1 vc (t ) 
each current is determined only by its voltage !

K p K s  3K 
K p K s  3K  
Ki s
  i 2 i2 0   i 2 i2 0 
 Kp  2
2
2 2  (s   0 )
2 2  (s   0 ) 
s  0

K p K s  3K  
Ks
2  K p K s  3K 
( a , b .c )
Gc ( s)     i 2 i 2 0
Kp  2 i 2
  i 2 i2 0 
3  2 2  (s   0 )
2 2  (s   0 ) 
s  0
K p K i s  3K i 0
 K p K i s  3K i 0

Ks
 
Kp  2 i 2 
 
2
2
2
2
2 2  (s   0 )
s  0
 2 2  (s   0 )

Marco Liserre



Ki s
0
0
K p  2 2

s  0


K
s


( a , b .c )
Gc1 (s)  
0
Kp  2 i 2
0

s  0


Ki s 

0
0
Kp  2 2

s   0 
[email protected]
Modulation and current/voltage control of the grid converter
Linear controllers : results (ideal grid conditions)
PI controller in a rotating frame
harmonic spectrum
current error
P+resonant controller for each phase
harmonic spectrum
Marco Liserre
current error
[email protected]
Modulation and current/voltage control of the grid converter
Linear controllers: results (equivalence of PI in dq and
P+res in )
PI controller in a rotating frame
P+resonant in stationary frame
Marco Liserre
triggering LCL-filter resonance
triggering LCL-filter resonance
[email protected]
Modulation and current/voltage control of the grid converter
Ac voltage control

When it is needed to control the ac voltage because the system should operate
in stand-alone mode, in a microgrid, or there are requirements on the voltage
quality a multiloop control can be adopted
The ac capacitor voltage
is controlled though the
ac converter current.
The current controlled
converter operates as a
current source to
charge/discharge the
capacitor.
Marco Liserre
L1
i
+
Vdc
+
-
+
Vc
V
i
CC
-
Vc
VC
V*c
[email protected]
Modulation and current/voltage control of the grid converter
Ac voltage control

The repetitive controller ensures precise tracking of the selected harmonics and
it provides the reference of the PI current controller. Controlling the voltage Vc’
the PV shunt converter is improved with the function of voltage dips
mitigation. In presence of a voltage dip the grid current Ig is forced by the
controller to have a sinusoidal waveform which is phase shifted by almost 90°
with respect to the corresponding grid voltage.
Vc’
-
Vref
FDFT(z)
+
Kf
Iref
+
E
Ig
Iload
z  Na
Ic’
PV converter
Vc’
Vref
+
Marco Liserre
-
Repetitive
control
Ic’
-
Iref
+
PI
[email protected]
Modulation and current/voltage control of the grid converter
Conclusions

The PR uses Generalized Integrators (GI) that are double integrators achieving very
high gain in a narrow frequency band centered on the resonant frequency and almost
null outside.

This makes the PR controller to act as a notch filter at the resonance frequency and
thus it can track a sinusoidal reference without having to increase the switching
frequency or adopting a high gain, as it is the case for the classical PI controller.

PI adopted in a rotating frame achieves similar results, it is equivalent to the use of
three PR’s one for each phase

Also single phase use of PI in a dq frame is feasible

Dead-beat controller can compensate current error in two samples but it is affected
by PWM limits and parameters mismatches
Marco Liserre
[email protected]
Modulation and current/voltage control of the grid converter
Bibliography
1.
D. G. Holmes and T. Lipo, Pulse Width Modulation for Power Converters, Principles and Practice. New York: IEEE Press, 2003.
2.
M. Kazmierkowski, R. Krishnan, and F. Blaabjerg, Control in Power Electronics – Selected Problems. Academic Press, 2002.
3.
X. Yuan, W. Merk, H. Stemmler, and J. Allmeling, “Stationary-frame generalized integrators for current control of active power
filters with zero steady-state error for current harmonics of concern under unbalanced and distorted operating conditions,” IEEE
Trans. on Industry Applications, vol. 38, no. 2, pp. 523–532, 2002.
4.
D. Zmood and D. G. Holmes, “Stationary frame current regulation of PWM inverters with zero steady-state error,” IEEE Trans. on
Power Electronics, vol. 18, no. 3, pp. 814–822, 2003.
5.
M. Bojrup, P. Karlsson, M. Alaküla, L. Gertmar, ”A Multiple Rotating Integrator Controller for Active Filters”, Proc. of EPE 1999,
CD-ROM.
6.
R. Teodorescu, F. Blaabjerg, M. Liserre and A. Poh Chiang Loh, “Proportional-Resonant Controllers and Filters for GridConnected Voltage-Source Converters” IEE Proceedings on Electric Power Applications.
7.
A. Timbus, M. Liserre, R. Teodorescu, P. Rodriguez, F. Blaabjerg, Evaluation of Current Controllers for Distributed Power
Generation Systems, IEEE Transactions on Power Electronics, March 2009, vol. 24, no. 3, pp. 654-664.
8.
R. A. Mastromauro, M. Liserre, A. Dell'Aquila, “Study of the Effects of Inductor Nonlinear Behavior on the Performance of Current
Controllers for Single-Phase PV Grid Converters”, IEEE Transactions on Industrial Electronics, May 2008, vol. 55, no 5, pp. 2043
– 2052.
9.
IEEE Std 1547-2003 "IEEE Standard for Interconnecting Distributed Resources with Electric Power Systems", 2003.
10.
IEEE Std 1547.1-2005 "IEEE Standard Conformance Test Procedures for Equipment Interconnecting Distribut ed Resources
with Electric Power Systems", 2005.
11.
IEC Standard 61727, “Characteristic of the utility interface for photovoltaic (PV) systems,”, 2002.
12.
IEC Standard 61400-21 “Wind turbine generator systems Part 21: measurements and assessment of power quality
characteristics of grid connected wind turbines”, 2002.
13.
IEC Standard 61000-4-7, “Electromagnetic Compatibility, General Guide on Harmonics and Interharmonics Measurements and
Instrumentation”, 1997.
14.
IEC Standard 61000-3-6, “Electromagnetic Compatibility, Assessment of Emission Limits for Distorting Loads in MV and HV
Power Systems”, 1996.
Marco Liserre
[email protected]
Modulation and current/voltage control of the grid converter
Acknowledgment
Part of the material is or was included in the present and/or past editions of the
“Industrial/Ph.D. Course in Power Electronics for Renewable Energy Systems – in theory and practice”
Speakers: R. Teodorescu, P. Rodriguez, M. Liserre, J. M. Guerrero,
Place: Aalborg University, Denmark
The course is held twice (May and November) every year
Marco Liserre
[email protected]