grid synchronization for power converters marco liserre [email protected] grid synchronization for...
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Marco Liserre [email protected]
Grid synchronization for power converters
Grid synchronization for power converters
Marco Liserre
Marco Liserre [email protected]
Grid synchronization for power converters
• Grid requirements for DG inverters• PLL Basics, PLL in power systems• Design of PLL• PLL for single-phase systems
– Methods to create the orthogonal component– Methods using adaptive filters
• PLL for three-phase systems• Conclusions• Reference papers
Outline
Marco Liserre [email protected]
Grid synchronization for power converters
Grid Distrurbances
Thomsen,1999; CIGRE WG14-31, 1999
Grid disturbances are not at all a new issue, and the utilities are aware of them. However, they have to take a new look because of the rapidly changing customers’ needs and the nature of loads (CIGRE WG14-31, 1999)
Marco Liserre [email protected]
Grid synchronization for power converters
Grid requirements for DG inverters
The following conditions should be met, with voltages in RMS and measured at the point of utility connection.
When the utility frequency is outside the range of +/- 1 Hz the inverter should cease to energize the utility line within 0.2 seconds.The PV system shall have an average lagging power factor greater than 0,9 when the output is greater than 50% rated.
Thus the grid voltage and frequency should be estimated and monitored fast and accurate enough in order to cope with the standard
Marco Liserre [email protected]
Grid synchronization for power converters
Grid synchronization requirements
A good synchronization of the current with the grid voltage is necessary as:
the standards require a high power factor (> 0.9) a ”clean” reference for the current is necesarry in order to cope with the harmonic requirements of grid standards and codes grid connection transients needs to be minimized in order not to trip the inverter
Distributed Generation systems of higher power have also requirements in terms of voltage support or reactive power injection capability and of frequency support or active power droop
Micro-grid distributed generation systems have wider range of voltage and frequency and the estimated grid voltage parameters are often involved in control loops
Marco Liserre [email protected]
Grid synchronization for power converters
Grid synchronization options and challenges
There are two basical synchronization methods: Filtered Zero Cross Detection (ZCD) PLL
Single-phase systems:The classical solution for single-phase systems was Filtered ZCD as for the PLL two orthogonal voltages are required. The trend now is to use the PLL technique also by creating ”virtual” orthogonal components using different techniques!
Three-phase systems: Three-phase PLL should deal with unbalnace hence with negative sequence Moreover in three-phase systems dynamics would be better if synchronizing to all three phase voltages, i.e. based on space vectors rather then on a scalar voltage
Marco Liserre [email protected]
Grid synchronization for power converters
Zero Cross Detection (ZCD) circuits
Resistive feedback hysteresis circuit
Dual point interpolation circuit
Dynamic hysteresis comparator circuit
Source: R.W. Wall, “Simple methods for detecting zero crossing,” IEEE IECON’03, pp. 2477-2481
Marco Liserre [email protected]
Grid synchronization for power converters
Filtered Zero Cross Detection (ZCD) based monitoring and synchronization
v
21x d t
T
1
2
T
f
V
s i n I
I
O V / U V
O F / U F
T R I P
F i l t e r
m a xV
m inV
m i nf
m a xf
R M S C A L C
2
x
R S T
ku
f i lvZ C D
m a xV
m i nV
m i nfm a xf
V
f
v f i lv
Filtering introduces delay. There are digital predictive FIR filters without delay bu with high complexity (very high order!)The RMS voltage and frequency are calculated once in a period poor detection of changes (sags, dips, etc.)
Marco Liserre [email protected]
Grid synchronization for power converters
-200
-100
0
100
200
v [
V]
Basic idea of synchronization based on a phase-locked loop:
Phase-locked technology is broadly used in military, aerospace, consumer electronics systems where some kind of feedback is used to synchronize some local periodic event with some recognizable external event
Many biological processes are synchronized to environmental events. Actually, most of us schedule our daily activities phase-locking timing information supplied by a clock.
A grid connected power converter should phase-lock its internal oscillator to the grid voltage (or current), i.e., an amplitude and phase coherent internal signal should be generated.
Event based synchronization(simple, discontinuous, …)
in
v
Phase-locked synchronization(continuous, predictive,…)
PLL basis
Marco Liserre [email protected]
Grid synchronization for power converters
Basic blocks:
Phase Detector (PD). This block generates an output signal proportional to the phase difference between its two input signals. Depending on the type of PD, high frequency ac components appear together the dc phase difference signal.
Loop Filter (LF). This block exhibits low pass characteristic and filters out the high frequency ac components from the PD output. Typically this is a 1-st order LPF or PI controller.
Voltage Controlled Oscillator (VCO). This block generates at its output an ac signal whose frequency varies respect a central frequency as a function of the input voltage.
Phase Detector
LoopFilter
VoltageControlledOscillator
fvvvdv
PLL basis
Marco Liserre [email protected]
Grid synchronization for power converters
PLL in power systems
va
vb
T1 T3
Evdc
T5
vc
ia
T4 T6 T2
LS
+-
LLRL
In 1968 Ainsworth proposed to use a voltage controlled oscillator (VCO) inside the control loop of a High Voltage Direct Current (HVDC) transmission system to deal with the novel, at that time, harmonic instability problem.
Subsequently, analog phase lockedloops (PLL) were proposed to be used as measurement blocks, which provide frequency adaptation in motor drives.
Marco Liserre [email protected]
Grid synchronization for power converters
Phase Locked Loop tuning
cos( )x
p ik k
ok
dk
c
esdv sin in inA t
PD LF
VCO
sin ωin in inv A t
cos ωVCO c outv t
Reference:
VCO output:
PD/Mixer output: sin ω cos ω sin sin2
dd d in in c out in c in out in c in out
Akv Ak t t t t
VCO angle: c o e out o et k s dt k s dt
if , then ,
Sm
all
sig
nal
an
alys
is: inωc sin 2 sin
2d
d in in out in out
Akv t
in out sin 22
dd in in in out
Akv t
The average value is 2
dd in out
Akv
sin in out in out if , then ,
Marco Liserre [email protected]
Grid synchronization for power converters
Phase Locked Loop tuning
2
( )( )
( )
pp
out i
pinp
i
kk s
s TH s
kss k s
T
;2p ip
ni
k Tk
T
1.8r
n
t
29.2;
2.3s
p is
tk T
t
11p
i
kT s
esdv
PD LF - HPI VCO
in outokmk
1
s
1 1o mk k assuming
that can be written as
2
2 2
( ) 2( )
( ) 2out n n
in n n
s sH s
s s s
with
4.6s
n
t
The PLL can be tuned as function of the damping and of the settling time
then
Marco Liserre [email protected]
Grid synchronization for power converters
The hold range H is the frequency range at which a PLL is able to maintain lock statically.
Key parameters of the PLL
(0)H o mk k LF
The lock range L is the frequency range within which a PLL locks within one-single beat note between the reference frequency and the output frequency.
For the PI, LF(0)=∞ and the hold range is only limited by the frequency range of the VCO
2 2 pL n
i
k
T
Pull-in time:
2L
n
T
The pull-in range P is the frequency range at which a PLL will always became locked, but the process can become rather slow. For the PI loop filter this range trends to infinite.
0 0.5 1 1.5 2 2.50
100
200
300
400
t [s]
[
rad
/s]
0 0.5 1 1.5 2 2.50
2
4
6
8
t [s]
[
rad
]
Lock-in time:
22
316in
Pn
T
Marco Liserre [email protected]
Grid synchronization for power converters
Phase Locked Loop: the need of the orthogonal component
11p
i
KsT
X
X
cos
sin
s
1
in
Vsin -in out
Vsin in int
Vcos in int
in outt
+++-
To eliminate the 2° harmonic oscillation from sin 2 sin2
din in out in out
Akt
and obtain it should be considered that sin2
din out
Ak
sin - sin cos cos sinin out in out in out
Marco Liserre [email protected]
Grid synchronization for power converters
Park transformation in the PD
cos( ) sin( )
sin( ) cos( )d out out
q out out
v v
v v
Park transformation:
sin( )
cos( )in
in
vV
v
sin cos cos sin sin
sin sin cos cos cosd in out in out in out
q in out in out in out
vV V
v
Assuming in=out :
sin
cosd in out
q in out
vV
v
11p
i
kT s
fvdv
LF VCO
1
sc
qv
dq
v
v
out
out
PD
inv Quadrature Signal
Generator
v
v
qv
dv
in
out
sin( )inv V v
Marco Liserre [email protected]
Grid synchronization for power converters
Park transformation in the PD
0
0int
d
q
v
02out
t
sin( ) ; 0in qv V v
0
0int
d
q
v
0
0outt
sin( ) ; 0in dv V v 11p
i
kT s
fvdv
LF VCO
1
sc
dq
v
v
out in
out
PD
inv Quadrature Signal
Generator
qv v
11p
i
kT s
fv
LF VCO
1
sc
qv
dq
v
v 2out in
out
PD
inv Quadrature Signal
Generator
dv v
PI on vd
PI on vq
From here on, it will be considered:
and PI on vq,, i.e.,
Therefore:
sinin inv v V 0qv
andout in dv v V
Marco Liserre [email protected]
Grid synchronization for power converters
Methods to create the orthogonal component
Transport Delay T/4
The transport delay block is easily implemented through the use of a first-in-first-out (FIFO) buffer, with size set to one fourth the number of samples contained in one cycle of the fundamental frequency.
This method works fine for fixed grid frequency. If the grid frequency is changing with for ex +/-1 Hz, then the PLL will produce an error
If input voltage consists of several frequency components, orthogonal signals generation will produce errors because each of the components should be delayed one fourth of its fundamental period.
11p
i
kT s
esdv
LF VCO
1
sc
qv
dqDelayT/4
v
v
PD
inv
inv
Marco Liserre [email protected]
Grid synchronization for power converters
Methods to create the orthogonal component Inverse Park Transformation
A single phase voltage (v) and an internally generated signal (v’) are used as inputs to a Park transformation block (αβ-dq). The d axis output of the Park transformation is used in a control loop to obtain phase and frequency information of the input signal.
v’ is obtained through the use of an inverse Park transformation, where the inputs are the d and q-axis outputs of the Park transformation (dq-αβ). fed through first-order low pass filters.
Although the algorithm of the PLL based on the inverse Park transformation is easily implemented, requiring only an inverse Park and two first-order low-pass filters
11p
i
kT s
esdv
LF VCO
1
sc
qv
dq
v
v
PD
inv
inv
dq
LPF
LPF
dv
qvv
v
Marco Liserre [email protected]
Grid synchronization for power converters
Methods to create the orthogonal component Second Order Generalized Integrator
2 2( ) ( )
d sS s s
f s
SOGI
d
q
f
2
2 2( ) ( )
qT s s
f s
2 2( ) ( )
v k sD s s
v s k s
2
2 2( ) ( )
qv kQ s s
v s k s
-60
-40
-20
0
20
Ma
gn
itud
e (
dB
)
10-1
100
101
102
103
104
-90
-45
0
45
90
Ph
ase
(d
eg
)
Frequency (Hz)
k=0.1k=1k=4
-60
-40
-20
0
20
Ma
gn
itud
e (
dB
)
10-1
100
101
102
103
104
-180
-135
-90
-45
0P
ha
se (
de
g)
Frequency (Hz)
k=0.1k=1k=4
( )D
( )Q
Marco Liserre [email protected]
Grid synchronization for power converters
k
k
outvvinv
cos
sin
OSCILLATOR
Methods using adaptive filters
Adaptive Notch Filter (ANF)2 2
2 2( ) ( )out
in
v sANF s s
v s ks
vout=0 when:
vout can not be directly used as PD in the PLL
t
vout=0 when:
vout can be used as PD in the PLL
int
koutv
vinv
cos
OSCILLATOR
cosin inv A t
Marco Liserre [email protected]
Grid synchronization for power converters
Methods using adaptive filters ANF-based PLL
PD
kv
cos
inv es
LF
VCO
1
scAdaptive Notch Filter
dvck
1
s
Very sensible to frequency variation ANF+PLL EPLL
More robust
Faster dynamic response
PDkv
cos
inv 11p
i
kT s
es
LF VCO
1
sc
sinAdaptive Notch Filter
dv
Conventional PLL structure
1
s
Combination of an ANF with a conventional PLL gives rise to the Enhanced PLL (EPLL)
Marco Liserre [email protected]
Grid synchronization for power converters
dv
kv v
PI
cos
ju u
v ( )V
ABPF
ff
v v’
sin
VCO
LF
PD
Enhanced PLL (EPLL)
Original structure of the EPLL
Methods using adaptive filters
×
K
×
×
90°
Kp
Ki sin
+ +
+
+
+-
y
A
Δω
ω0
θ
BPAF LP VCO
v e
1s
1s
1s
Marco Liserre [email protected]
Grid synchronization for power converters
2 2( ) ( )
v k sD s s
v s k s
SOGI-PLL
Methods using adaptive filters
2 2( ) 1 ( ) ( )
v ksABPF s ANF s s
v s ks
Adaptive band-pass filter:
Damping factor is a function of the detected frequency value
Second order generalized integrator follower:
If ’ can change, SOGI follower can be seen as an adaptive band-pass filter with damping factor set by k and unitary gain
As in the EPLL, a standard PLL can be used to detect grid frequency and angle
ju is 90º-leading v’ when the PLL is synchronized in steady state
ju=-qu and qu qv’
It seems intuitive to use -qu (instead ju) as the feedback signal for the PD of the PLL
v
VCO
kv v
qv
PI
juffsin
LF
SOGI
v
PD
Conventional PLL structure
Marco Liserre [email protected]
Grid synchronization for power converters
SOGI-based Frequency Locked Loop (SOGI-FLL)
Methods using adaptive filters
vk
v v
qv
SOGI
1
ff
v
qv
FLL
Does not need any trigonometric function since neither synchronous reference frame nor voltage controlled oscillator are used in its algorithm.
Is frequency-adaptive by using a FLL and not a PLL.
Is highly robust in front of transient events since grid frequency is more stable than voltage phase-angle.
Attenuates high-order harmonics of the grid voltage.
Entails light computational burden, using only five integrators for detection of both sequence components.
Marco Liserre [email protected]
Grid synchronization for power converters
Distorted and unbalanced voltage vector
Three-phase grid synchronization
tt
1SV
1SV
SV
a
b
c
1SV
1SV
11 SS VV11 SS VV
1 2 1 2 1 1 1( ) ( ) 2 cos( 2 )S S S S SV V V V t v
1 11
1 1 1
sin( 2 )tan
cos( 2 )S
S S
V tt
V V t
a
b
c
SV
1SV
5SV
5SV
1SV
v S S Sn
S SnV V V V n t 1 2 2 12 1cos
tV n t
V V n tSn
S Sn
ta ns in
cos1
1
1
1
Neither constant amplitude nor rotation speed
Marco Liserre [email protected]
Grid synchronization for power converters
Characterization of voltage dips
0 0.02 0.04 0.06 0.08 0.1-1.5
-1
-0.5
0
0.5
1
1.5
V=0.5<-20 ;F=0.75<-40 V+=0.61589<-32.0197 ;V-=0.16411<108.5995
t [s]
v abc [p
u]
0 0.02 0.04 0.06 0.08 0.1-1.5
-1
-0.5
0
0.5
1
1.5
V=0.5<-20 ;F=0.75<-40 V+=0.61589<-32.0197 ;V-=0.16411<108.5995
t [s]
v [p
u]
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
V=0.5<-20 ;F=0.75<-40 V+=0.61589<-32.0197 ;V-=0.16411<108.5995
v [pu]
v [p
u]
312 2
312 2
Type C
Sa
Sb
Sb
V F
V F jV
V F jV
31
2 2
312 2
Type D
Sa
Sb
Sb
V V
V V jF
V V jF
1 12
1 12
Type C
S
S
V V F
V V F
1 12
1 12
Type D
S
S
V V F
V V F
Phase-voltages from characteristic parameters
Sequence components from characteristic parameters
Marco Liserre [email protected]
Grid synchronization for power converters
Three-phase Synchronous Reference Frame PLL
Three-phase grid synchronization
PIs1
SavSbvScv
dqT
Sdv
Sqv
ˆSd Sv v
-150
-100
-50
0
50
100
150
0
1
2
3
4
5
6
7
0 25 50 75 100-50
0
50
100
150
t [ms]
-150
-100
-50
0
50
100
150
0
1
2
3
4
5
6
7
0 25 50 75 100-50
0
50
100
150
t [ms]
Balanced voltage
Unbalanced voltage
Sv
Sv
ˆ t
t
Sd Sv v
0Sqv
Sd Sv v
0Sqv
11 1
1( )
ˆ ˆcos( ) cos( )
ˆ ˆsin( ) sin( )
Sd
S S Sdq Sq
v t tV V
v t t
v
Marco Liserre [email protected]
Grid synchronization for power converters
-150
-100
-50
0
50
100
150
0
1
2
3
4
5
6
7
-50
0
50
100
150
Three-phase Synchronous Reference Frame PLL
Three-phase grid synchronization
The SRF is not able to track instantaneous evolution of the voltage vector when the PLL bandwidth is low
Sv
ˆ t
Sdv
Sqv
0 25 50 75 100-150
-100
-50
0
50
100
150
t [ms]
t 1ˆ Sv
1 1
( )
1 cos( 2 )v
' sin( 2 )S S Sdq
tV V
t t
' t Near of synchronization:
sin( ') 't t cos( ') 1t ' 2t t PI
s1
SavSbvScv
dqT
Sdv
Sqv
ˆSd Sv v
1
1 11sin(2 ) ' 'S
Sq S SS
Vv V t t V
V
1
1sin(2 )S
S
Vt t
V
i
p
kk
s
1
s1
SV *
1Sqv
2
2 2
ˆ 2( ) ( )
2c c
c c
sP s s
s s
1c S iV k
1
2p S
i
k V
k
Marco Liserre [email protected]
Grid synchronization for power converters
-150
-100
-50
0
50
100
150
Three-phase Synchronous Reference Frame PLL
Three-phase grid synchronization
Setting a low PLL bandwidth and using a low-pass filter it is possible to obtain a reasonable approximation of the positive sequence voltage but the dynamic is too slow.
Sv
0
1
2
3
4
5
6
7
-50
0
50
100
150
Sqv
Sdv
0 25 50 75 100-150
-100
-50
0
50
100
150
t [ms]
1ˆ Sv
PIs
1
SavSbvScv
dqT
Sdv
Sqv
ˆSd Sv v
Repetitive controller
Advanced filtering strategies can be used to cancel out the double frequency oscillation keeping high locking dynamics, e.g., a repetitive controller based on a DFT algorithm. Additional improvements are added to these filters to make them frequency adaptive.
Marco Liserre [email protected]
Grid synchronization for power converters
Decoupled Doubled SRF-PLL. Decoupling
Three-phase grid synchronization
1
11 1
11 1
1( )( )
ˆ ˆcos( ) cos( )
ˆ ˆsin( ) sin( )
SdS S S Sdqdq Sq
v t tT V V
v t t
v v
1
11 1
11 1
1( )( )
ˆ ˆcos( ) cos( )
ˆ ˆsin( ) sin( )
SdS S S Sdqdq Sq
v t tT V V
v t t
v v
' t Near of synchronization:
1
11 1
1( )
1 cos( 2 )ˆ sin( 2 )
S S Sdq
tV V
tt
v
1
11 1
1( )
cos(2 ) cos( )
sin(2 ) sin( )S S Sdq
tV V
t
v
cos(( ) ) sin(( ) )cos( )cos( ) sin( )
sin(( ) ) cos(( ) )sin( )
n
n
n nSd m m m mS
S Sn nSq S
v n m t n m tVV V
v n m t n m tV
cos(( ) ) sin(( ) )cos( )cos( ) sin( ) .
sin(( ) ) cos(( ) )sin( )
m
m
m mSd n n n nS
S Sm mSq S
v n m t n m tVV V
v n m t n m tV
Generic decoupling cell:
cos
nSdv
sin
mSdv mSq
v
nSqv
*nSd
v
*nSq
v
m
nDC
nd
nq
mdmq *nd
*nq
n-m
This terms act as interferences on
the SRF dqn rotating at n frequency and
viceversa
Marco Liserre [email protected]
Grid synchronization for power converters
Three-phase grid synchronization
y
.
Decoupled Doubled SRF-PLL
1Sdv
1Sqv
1d1q
1
1DC
1d1q
*1d
*1q
1Sdv
1Sqv
*1Sd
v
*1Sq
v
*1Sd
v
*1Sq
v
1Sdv
1Sqv
1
1ˆ SSdv
v
1Sqv
T Sv
1dqT
1dqT
abcSv
1d 1q
1
1DC
1d1q
*1d
*1q
ip kk
LPF
LPF
LPF
LPF
1
*
Sqv
*1Sq
v
2 2q d qv v v
f
f
PLL input normalization
Marco Liserre [email protected]
Grid synchronization for power converters
Conclusions
PLL is a very useful method that enable the grid inverters to: Create a "clean" current reference synchronized with the grid Comply with the grid monitoring standards
The PLL generate is able to track the frequency and phase of the input signal in a designed settling time
By setting a higher settling time a "filtering" effect can be achieved in order to obtain a "clean" reference even with a polluted grid.
Some PLLs need two signals in quadrature at the input. For single-phase systems as there is only one signal available, the
orthogonal signal needs to be created artificially. Transport Delay, Inverse Park Transformation, or Second Order
Generalized Integrators are some the methods used for quadrature signal generation.
Adaptive notch filters canceling fundamental utility frequency are used as phase detectors in PLLs
FLL based on a SOGI is a very effective method for single phase synchronization
Marco Liserre [email protected]
Grid synchronization for power converters
References1. J. D. Ainsworth, “The phase-locked oscillator-a new control system for controlled static
convertors,” IEEE Transactions on Power Apparatus and Systems, vol. 87, no. 3, pp. 859-865,
Mar. 1968.
2. G. C. Hsieh, J. C. Hung, Phase-locked loop techniques – A survey, IEEE Trans. On Ind.
Electronics, vol.43, pp.609-615, Dec.1996.
3. F. M. Gardner, Phase Lock Techniques. New York: Wiley, 1979.
4. L. D. Zhang, M. H. J. Bollen Characteristic of voltage dips (sags) in power systems, IEEE Trans.
Power Delivery, vol.15, pp.827-832, April 2000.
5. F. Blaabjerg, R. Teodorescu, M. Liserre, and A. V. Timbus, “Overview of Control and Grid
Synchronization for Distributed Power Generation Systems”, IEEE Trans. on Ind. Electronics, Vol.
53, Oct. 2006 Page(s):1398 – 1409
6. M. K. Ghartemani, M.R. Iravani, “A method for synchronization of power electronic converters in
polluted and variable-frequency environments,” IEEE Trans. Power Systems, vol. 19, pp. 1263-
1270, Aug. 2004.
7. M.K. Ghartemani, M.R. Iravani, “A Method for Synchronization of Power Electronic Converters in
Polluted and Variable-Frequency Environments,” IEEE Trans. Power Systems, vol. 19, Aug.
2004, pp. 1263-1270.
8. H.-S. Song and K. Nam, “Dual current control scheme for PWM converter under unbalanced input
voltage conditions,” IEEE Trans. On Industrial Electronics, vol. 46, no. 5, pp. 953–959, 1999.
Marco Liserre [email protected]
Grid synchronization for power converters
References1. P. Rodríguez, A. Luna, I. Candela, R. Teodorescu, and F. Blaabjerg, “Grid Synchronization
of Power Converters using Multiple Second Order Generalized Integrators,” IECON’08, Nov.
2008.
2. P. Rodríguez, J. Pou, J. Bergas, J.I. Candela, R. Burgos and D. Boroyevich, “Decoupled
Double Synchronous Reference Frame PLL for Power Converters Control,” IEEE Trans. on
Power Electronics, March 2007.
3. P. Rodriguez, R. Teodorescu, R.; I. Candela, I.; A.V. Timbus, M. Liserre, F. Blaabjerg, “New
Positive-sequence Voltage Detector for Grid Synchronization of Power Converters under
Faulty Grid Conditions,” PESC '06, June 2006.
4. M Ciubotaru, Teodorescu, R., Blaabjerg, F., “A New Single-Phase PLL Structure Based on
Second Order Generalized Integrator”, PESC’06, June 2006.
5. P. Rodríguez, A. Luna, M. Ciobotaru, R. Teodorescu, and F. Blaabjerg, “Advanced Grid
Synchronization System for Power Converters under Unbalanced and Distorted Operating
Conditions,” IECON’06, Nov. 2006.
6. S.-K. Chung, “Phase-Locked Loop for grid-connected three-phase power conversion
systems,” IEE Proceedings on Electronic Power Applications, vol. 147, no. 3, pp. 213–219,
2000.
7. Francisco Daniel Freijedo Fernández, “Contributions to Grid-Synchronization Techniques for
Power Electronic Converters”, PhD Thesis, Vigo University, Spain, 2009
Marco Liserre [email protected]
Grid synchronization for power converters
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